Caracter scored 20 points and pulled down 13 rebounds, and UTEP snapped the Blazers' seven-game winning streak with a 74-65 victory on Saturday night.
"We just had to stay positive and tell ourselves that this was our game," said Caracter.
With the score tied at 58 at the end of regulation, an apparent winning shot by Julyan Stone was nullified when teammate Arnett Moultrie was called for basket interference.
Each team managed only five points in the first overtime before Caracter had five points himself to help the Miners outscore the Blazers 11-2 in the second.
Jeremy Williams added 16 points for UTEP (15-5, 6-1 Conference USA), Randy Culpepper had 14 and Christian Polk 11. The win leaves the Miners tied with UAB and Tulsa atop the conference standings.
"I'm so proud of my kids, the way we toughed out a win against a team in first place, a Top 25 team. For us to come in here and hold them to 28 percent shooting on their homecourt, that's a great defensive effort," said Miners' head coach Tony Barbee.
Barbee said his players won the game because they kept their "poise, composure and confidence" but admitted he lost his when he was hit with a technical foul for arguing with the referees during a timeout late in the second half.
Kenneth Cooper had 18 points and 12 rebounds, Aaron Johnson scored 15 points and Howard Crawford added 10 for UAB (18-3, 6-1).
There was a scary moment with 2:41 left in regulation when Johnson and Culpepper collided heads and both players fell to the floor. Johnson left the court and did not return. UAB officials said he had a broken nose.
UAB Coach Mike Davis said losing the 5-foot-8 guard down the stretch hurt the Blazers.
"He's the only guy who can really run our offense. He would have had his hand on the basketball late in the game. He would have been the guy getting the ball to other people," Davis said.
Davis said he's not sure how much time Johnson will miss because of the injury.
Culpepper was able to stay in the game.
It was UTEP's first victory against a ranked opponent since 2001 and its first road win against a Top 25 team since 1993. The Miners came in having lost 13 in a row against ranked opponents.
The Blazers shot only 26 percent from the field in the first half and trailed 33-26 at the break. UTEP shot 48 percent in the first 20 minutes and led by as many as nine late in the half.
The Blazers slowly fought back in the second half, pulling within 1 on back-to-back 3-pointers by Jamarr Sanders before Cooper's dunk gave them a 48-47 lead with seven minutes to play.
Polk hit several big shots for UTEP in the second half, including a 3-pointer to tie the game at 58 with 1:35 left in regulation.
"It's the first time I can say we didn't take advantage of our opportunities. We had 22 offensive rebounds and 12 second-chance points. We just didn't take advantage of it," Davis said.