Nice twist at the end!  I wasn't expecting it, though I thought something was up when you started talking about "cultures"   I enjoyed it for the most part, but there were some logical flaws and pedantic things that need fixin'

You should go back and spell-check/grammar-check the entire piece, though.  I'd suggest doing it manually, as well.  It'll only take a few minutes, and it will improve the piece immeasurably.

I don't remember how it all began, don't know the cause of their dispute, but somebody mounted the first attack; us, them, or somebody else. But within a month, it was all out war. Countless soldiers, tanks, aircraft, warships, bombing runs and blitzkrieg, innocents dying in the masses. But it all wasn't enough for them. They had big plans, and feared the big boom. So they picked me and a few thousand others at random, stuck us into these self-contained bunkers, and then continued their war like we never existed.

There are too many "but"s in this section.  Cut a few of them out!

There seems to be a bit of confusion in regards to who exactly is going to be working on the creation process.  At times, there are "scientists"

Find? Create! This bunker has the facilities, and enough scientists. We should make it. No, we will!

But in other places, the main character is apparently the only one who's qualified to work on it:

Maybe I can not begin creation yet, but I certainly can begin design.

I was free to conduct my experiments.

Which is it?

Sometimes you give out explanation when you don't need to:

but there was one who actually tried to break into the weapons storage and go on a killing spree

It should be fairly obvious why someone would break into the weapons storage unit.  You can cut the "Killing spree" part out.

Just in case, we picked seven council members at random, and each secretly picked a digit to create the new security code for the weapons storage. That way, no single person knows it.

"Just in case" is a bit strange considering the previous sentence.  The last sentence here is another example of needless explanation.

The process of "sealing" is really vague.  The bunker is permanently sealed, yet they've been outside and back in again.  The weapons storage is permanently sealed but it gets opened later on.  The lab is on a "timed seal" which later becomes a "permanent seal."  This last seal not only stops people getting in, but appears to be air-tight? 

In an impromptu council meeting, it was decided to remove the insufficient unit, cut one or two sectors permanently off, and redistribute their population to other sectors.

If the unit just can't handle the amounts of power, removing it shouldn't be necessary if you cut the amount of power that's going to it.

The council announced our plans today. Needless to say, our people were not exactly thrilled. Nobody wants to be cut of the power supply and relocated. Tomorrow's council meeting was canceled.

Why was the meeting canceled?  "cut of the power supply" should be reworded, as well.

The number is ofsystem failures is steadily on the rise.

This sentence confused me a lot until I realised it should say "of".

Access to the database was sluggish at first, but after fixing a few components, operating parameters have been restored to ideal levels

Too vague.  Operating parameters for what?  Access to which database?  How can you fix components with a database?  Or were they on a machine, and those got fixed?

All specifications have been entered according to the required protocol, but the estimated completion time is a week nonetheless. As a safety measure against sabotage, I've sealed the room with a timed lock.

Here's where my confusion about the sealing came in.  After re-reading, I see that it's only sealed for a week at first, and then it opens.  Maybe you could change it to "...I've sealed the room until completion."

My god! What have we done? What have we unleashed? It's happened now. It's happening all over again. War... Others might call it a riot, but I call it war. They've broken the seals on the weapons storage. Sector has turned upon sector, brother on brother. Charred corpses are littering the corridors, screams of agony and rage sound through the halls. As far as I can tell, the council's dead.

This doesn't even make sense.  I thought it was that his creation escaped, killing everybody.  Instead it seems like somebody got access to the "permanently sealed" weapons unit and went berserk.

Problems with that:

Why "What have we done?" and "What have we unleashed?"  As far as I can tell, they haven't unleashed anything.  Everybody just went berserk for no reason.  Are they supposed to have gone berserk because of the experiments?  That isn't made clear. 

If the "Seals" were permanent, how were they able to break them so easily (with no weapons)?

Using various pieces of laboratory equipment, we've put together a makeshift generator. It won't last forever, but it should get us through the riots. It's even strong enough to power the main database and all other equipment necessary for our experiments, so we have decided to begin our work. It is better than wasting our days with worry until we go insane.

More vagueness with "database".  Is it just being used to sound sciencey?  Also, I thought they'd already started the experiments a week ago?  Was that just set-up time?  It sounded like they'd started.  Here we go back to the "We vs I" thing again with scientists.

The injection itself was successful, but we had a tragic incident. Just as we were placing the last cell colonies in the last vats, we heard the sound of shattering glass from the next room, then gunshots. Loud, panicked shouts, accusations and courses. What are we doing this for? What's the point of all this? We should save ourselves, not a failed civilization.

I turned a deaf ear to the shouts, but hurried over to the next room to help. But when I arrived, it was already too late. I'm all alone now.

Who is "We" at the beginning?  If the other scientists somehow killed each other with one weapon, who else "hear the sound of shattering glass, then gunshots."?  Then we switch over to "I" as well.  If he's hurrying over to the next room to help, how was he turning a deaf ear?  Or were those shouts coming from outside the lab from angry rioters?  If that's the case, how is the room still "sealed", as it's mentioned being later on?

Oh, look at my children! Look at my creation! Ain't they sweet? Growing so swiftly, and so perfectly designed. Ohhh, the power... I hold the raw power of life in my hands. When they are born, I shall be their god. Their god, I say!

This is way too out of character.  I know it's supposed to be, but it just didn't work for me, even then.

Why were they so aggressive.

Missing a question mark.

 I will fill all the nutrient vats, not just the ones in this laboratory, but also the ones down in the sub-basement, a place I only learned off today in the records, with row upon row of nutrient vats, thousands of them, with the seed of creation. It will be my last act, for I have to venture out into the contaminated corridors to do so. But it shall be worth it.

This is pretty random.  The chances of there being a "sub-basement" which just happens to have all these nutrient vats is pretty far-fetched!  Even if there were, would it have escaped notice of the rioters?

Despite all the logical flaws and not-100%-thought out stuff inherent to most first drafts, I liked the premise and enjoyed it!  The character was fairly easy to relate to (everybody likes to pretend they're the logical sane one, after all) and I think you did a fairly good job (logic aside) of detailing how the society falls apart.

Looking forward to a second draft!

Ha ha, that was well written, it was actually fun to read, which is something critical for a short story. The rhythm is dead on, perfect timing and delivery. I love the short paragraphs of information, just perfect. Also I love how you manage to drop small clues but avoid saying too much, very well written.

The only part that was a bit disappointing was the beginning.  I wish you would have described the war a bit more, try looking up fallout intro on youtube the intro narration they have for that video game is perfect for a war setting. Try to capture the desolation of their situation, their despair.

Also why were they selected randomly? Why aren't they the smartest or most powerful of their race? Meh maybe I'm too nitpicky.

Anyhow good job here, a minor polish will make this story a great piece.