Vanderbilt (14-10, 6-6 SEC), led by James Siakam's 16 points, came back to beat Texas A&M 57-54 in OT after trailing by 11 in the second half. Damian Jones' three-point play a minute into extra time was the first lead for the Commodores, 52-49.
Odom sent the game into overtime at 49-49 with his long jumper.
Referees reviewed Odom's shot to see if it was a 3-point winner, but ruled it a tying basket.
Siakam's dunk had cut it to 49-47 with 9 seconds left, and Dai-Jon Parker forced a jump ball with the possession favoring Vanderbilt to set up Odom's game-tying basket.
"I had a rough game, but my shot didn't feel bad all game, so I figured by the law of averages I was due to make one," said Odom, who fouled out in OT.
"I had missed one at the end of (last week's loss to) Arkansas that I had been thinking about a while. It was a perfect opportunity to rectify that."
Texas A&M (14-11, 5-7) was down 55-52 with a chance to tie it, but Jamal Jones was called for an offensive flagrant foul against Carter Josephs with 8.1 seconds left.
Luke Kornet hit 1 of 2 free throws for a 56-52 lead after Josephs had to leave for stitches following the flagrant. Kyle Fuller then sank another free throw to preserve the win.
Siakam went 12 of 14 from the foul line and had 10 rebounds.
"I'm very proud of our team," Vanderbilt coach Kevin Stallings said. "On a day when we were 3-of-26 from 3 and shot 31 percent, on a day when it felt like we couldn't get anything going offensively, we found a way to get a victory. We had a number of players who made big plays."
Jamal Jones led the Aggies with 17 points, including 5 of 7 from long range, to go along with 13 rebounds. Kourtney Roberson added 11 points.
A&M hit 9 of 20 free throws and missed 2 of 3 in the final minute of regulation before Jamal Jones and Shawn Smith each converted a pair with less than 30 seconds left.
"We've got to make free throws, and that I thought that cost us the game," Aggies coach Billy Kennedy said. "And then I don't know what happened at the end, the jump ball call on the foul."
Kennedy, coaching on the other baseline, said Vanderbilt's unique end-zone benches at Memorial Gym give the Commodores "the best home-court advantage I've ever played against."
"You want to call a timeout in that situation and they're calling 'Jump ball, jump ball, jump ball' to the refs," Kennedy said. "And we're calling 'Foul, foul, foul.' They hear them a lot easier than they hear us."
Parker had urged Fuller the play before to go for a tie-up if trying to foul since VU had the possession arrow.
"He just told me to do that, and he went out there and did it," Fuller said.
Vanderbilt gained a 55-50 lead in overtime after Siakam sank two free throws, but Alex Caruso cut it to 55-52 on his foul shots for A&M.
"We just kept telling each other we have to keep this lead, we have to keep getting stops," Fuller said. "Damian was getting some big rebounds, some man rebounds. It was real good to see that."
The Commodores are down to seven scholarship players and lost at South Carolina just two days ago, after a one-day snow delay.
The Aggies scored the first 10 points of the game and held their biggest lead at 44-33 on Jordan Green's 3-pointer. Vanderbilt used a 7-0 run to close to 44-40 on Siakam's free throws with 1:19 left.
Texas A&M scored the first 10 points of the game on a pair of Roberson buckets and back-to-back 3-pointers by Antwan Space and Jamal Jones. Damian Jones broke the 10-0 spurt with a basket inside.
The Aggies didn't score for nearly 9 minutes in the first half after Caruso's 3-pointer made it 15-5, but led 23-19 at halftime.