Abrams, who played all 45 minutes in the high altitude, scored 29 points including five in overtime to carry the Longhorns.
With the game tied 72-72 with 1.6 second to play, James walked to the free throw line for two needing only one to secure the Longhorns 17th victory of the year. The junior missed both and the game went to overtime.
Sensing a chance for redemption in quick turnaround fashion, James dominated the overtime period scoring six points and pulling down six of his game high 14 rebounds. The spurt included two points and three rebounds in the last 20 seconds.
With 40 seconds remaining in regulation it appeared as if the Longhorns were going to earn a hard fought victory in regulation as freshman Varez Ward calmly walked to the line and sank a pair of free throws to give the Longhorns (17-7, 6-4) a 72-70 lead.
After Higgins missed a jumper, Colorado (9-15, 1-9) senior forward Jermyl Jackson-Wilson snatched an offensive rebound. After a Colorado timeout, Jackson-Wilson drove the ball down the middle of the lane and was fouled. The 57% free throw shooter knocked down both to tie the game before James drove the length of the court and was fouled.
The Longhorns led 36-32 at the half. Texas appeared in control of the game running out to a 30-15 lead with 5:31 to play in the half.
Higgins led the Buffalos with 34 points and nine rebounds. Thorne III and Wilson joined the sophomore in double figures with 10 apiece.
Abrams led the Longhorns with 29 points, while James added 21 points and 14 rebounds.
A Look at the Numbers
The Longhorns shot 57.4% (31-54) overall, 40.0% (6-15) from three and 60.7% (17-28) from the free throw line.
The Buffalos converted 48.9% (23-47) from the floor, 35.7% (5-14) from the three point line and 86.2% (25-29) from the foul line.
The Longhorns won the battle of the boards 29-27 and the turnover battle 16-12.
A Look at the Big 12
Texas is now 6-4, while Kansas State 6-5 and Nebraska at 5-5. For Texas to get a first round bye in the Big 12 Tournament, they must finish a game ahead of both Kansas State and Nebraksa due to losses head to head.
Texas will now face the quick turnaround "Big Monday" game at Texas A&M. A win by the Longhorns on Monday would be the first in four years in College Station. The Longhorns have lost the last two meetings at Texas A&M by an average of 17.5 points per game.
The Aggies will also be facing the quick turnaround, but have had the week off before tonight's game against the Baylor Bears in Waco.