The 2001 Game of the Year is Halo: Combat Evolved.
The runner up is Grand Theft Auto III.
The master thread for all of these contests can be found here.

We are going to have to do a runoff.  Four games ended up with far more votes than the others and then on top of that the "Other" category got a whole lot of votes.  There were 13 different games mentioned in the comments as getting an "Other" vote, but only one of those games got more than one.  It will be joining the other top 4.  (The other 12 games are worthy candidates too, but there aren't enough poll slots to include them all, so I'm only including the one that got more than one vote.)

First here are the 4 games that made the cutoff: 

Final Fantasy X (PS2)
Final Fantasy gets a serious upgrade.   Just a year after the release of the PS2 Squaresoft would release this epic RPG with fantastic looking graphics and almost all of the dialogue completely voice acted.  Not only did this game look and sound better than most games even near the end of the PS2’s life cycle, but Final Fantasy X still holds up pretty well today, 22 years later.  Final Fantasy X would sell over 8.5 million copies on the PS2 alone, and the Final Fantasy X series has sold over 21.1 million copies (with most later versions bundling X and X-2 together).

Grand Theft Auto 3 (PS2)
The game that made Grand Theft Auto a household name.  GTA 3 introduced most gamers at the time to 3D open-world action gaming, a type of gameplay not possible before Generation 6.  GTA 3 is rated as 97 on Metacritic and sold over 8.3 million units on the PS2 alone.  Across all platforms it has sold over 14.5 million copies.

Super Smash Bros Melee (Gamecube)
Smash Bros explodes in popularity with this second entry.  The game created a huge following in competitive gaming tournaments and even after later Smash Bros entries were released, the tournament community would often return to Melee as their go to game.  Super Smash Bros Melee would sell over 7.4 million units making it the best selling Gamecube game, and the first time a non-Mario game was the best seller on a Nintendo home system.

Halo: Combat Evolved (Xbox)
Microsoft launches both their first console and their flagship game series at the same time with Halo: Combat Evolved.  Halo set a new standard for FPS games going forward, receiving a 97 rating on Metacritic.  It has sold over 6 million copies total including 5 million copies on the original Xbox.

And now the one "Other" game that will be joining them:

Civilization III (PC/Mac)
Sid Meier hands the reigns over to Jeff Briggs and Soren Johnson for this third entry in the series, and the game still proves to be extremely fun and popular.  Civ III scored a 90 on Metacritic.

Runoff voting for 2001 Game of the Year will end on Tuesday, Oct 31 at 11:59 PM, EST.

Games that didn't make the cut to the Runoff round:
DDRMAX
Black and White
Gran Tourismo 3: A-Spec
Metal Gear Solid 2: Sons of Liberty
Animal Crossing

Here are some other notable games from the year 2001:

Tekken 4 (Arcade)
Civilization III (PC/Mac)
IL-2 Sturmovik (PC)
Harry Potter and the Philospher’s/Sorcerer’s Stone (Playstation)
Conker’s Bad Fur Day (N64)
Shenmue II (Dreamcast)
Crash Bandicoot: The Wrath of the Codex (PS2)
Devil May Cry (PS2)
Dynasty Warriors 3 (PS2)
Ico (PS2)
Jak and Daxter: The Precursor Legacy (PS2)
Max Payne (PS2/PC)
Onimusha: Warlords (PS2)
Silent Hill 2 (PS2/XBox)
The Simpsons Road Rage (PS2/XBox/Gamecube)
SSX Tricky (PS2/Xbox/Gamecube)
Tony Hawk’s Pro Skater 3 (Playstation/PS2/GBC/Gamecube)
Luigi’s Mansion (Gamecube)
Pikmin (Gamecube)
Dead or Alive 3 (XBox)
Project Gotham Racing (XBox)
Advance Wars (GBA)
Golden Sun (GBA)
Mario Kart Super Circuit (GBA)
Mega Man Battle Network (GBA)
Wario Land 4 (GBA)

Halo: Combat Evolved.

It's one of those games where it's hard to overstate how much of a breakthrough moment it was if you weren't there.

The scale, the graphics, the music, the AI, it was on a whole other level from any other shooter yet made to that point. It revolutionized the genre to the point where if you pick almost any current gen FPS game more than two decades later, it will owe much of its DNA to Halo. What Mario did for first 2D then 3D games, Halo did for first person shooters.

It practically single handedly carved a place for Xbox in the console space, and set the tone for much of the 2000s as a decade. But most importantly, it was just a blast to play.

Thera are very few games that had such an enormous impact on gaming and also on society to some extent than GTA 3. On top of that it was a damn fine game and a technical miracle for its day and age. So I'll go with that.

Mega Man Battle Network for me. Also loved THPS3 though. I feel like Silent Hill 2 should have been on the poll.

I have to go with Sons of Liberty, though in some ways I might consider Melee a better game in a vacuum, it just doesn't have as much draw for me after Brawl and Ultimate that surpass it in my eyes. My top 5 for the year would be:

1. Sons of Liberty
2. SSB Melee
3. Silent Hill 2
4. Final Fantasy X
5. Luigi's Mansion/Pikmin (can't decide)

Also have to give shoutouts to Golden Sun.

Halo I actually played for the first time recently. Mostly pretty fun, especially the vehicles, but the checkpoints are a travesty.

Devil May Cry.

This year is imo by far the best ever for an individual platform. The PS2 had an insane amount of great games in 2001, some quite influential, some started sub-genres (DMC, ICO, GTA3), and some are of the biggest sellers of the generation (GTA3, GT3, FFX, and MGS2).

A few great games not mentioned here include Ring of Red, Klonoa 2, Maximo, Baldur's Gate Dark Alliance, Twisted Metal Black, Ace Combat 4, and Virtua Fighter 4. The "not mentioned" list alone can take on entire years!

I'm surprised Sonic Adventure 2 isn't mentioned here. Especially, when it's a nice swan song for the Dreamcast.

This is tough year to vote, no matter how I look at it.

My favourite game from that year, by far, is Gothic. You know that 3rd person perspective, open world narrative RPG most people keep hearing about, that influenced so many developers (including Witcher devs), draws its influence from later "world simulation" Ultimas and has cult following, especially in Europe, yet not many people actually played it. Yeah, that one.
There was Arcanum: Of Steamworks and Magick Obscura. From the devs that created Fallout, but this time with a world that's even more reactive to you. I really liked it, but Steampunk is not my thing, so while mechanically better game than Fallout 1/2, it never rise to that status for me as FOs did.
Then there's Max Payne. Oh man, what a great action-adventure game with story and atmosphere soaking in noir genre, that introduced bullet time as a major gameplay concept.

But this is the year that has GTA III and Halo: CE. And while I don't generally like to vote for games that are sequels or refinements of previous ideas, these two were massive.
Eventually, I have to go with GTA III over Halo: CE, since FPS genre was not only well established, but had Half-Life happen just 2 years before that single-handedly changed (for better or worse) how FPS games were made from that point.

Other games I really liked form that year are Conker's Bad Fur Day, ICO, Return to Castle Wolfenstein, Black and White, Wizardry 8 and special mention goes to Severance: Blade of Darkness - Souls before Souls of a sorts, since it didn't have respawn mechanism, but overall gameplay and level design reminds of Souls, just with much more satisfying combat.

Halo for me and it isn't close. I still play the original to this day. I never liked GTA.