Singletary finished with 43 points to break the tournament scoring record of 38 set by Iowa State's Marcus Fizer in 2000 against Baylor.
"I was just in the zone, I think," Singletary said. "We didn't want to lose this game. We didn't want to go home."
Texas Tech advances to play No. 14 Missouri on Thursday.
The last basket in Singletary's amazing stretch was a driving layup past Bryan Davis that put the Red Raiders (14-18) up 79-78 with 39.4 seconds left and erased the last part of a 21-point deficit. Alan Voskuil and John Roberson followed with a pair of free throws apiece to push the lead to five.
Singletary, a sophomore whose previous scoring best was 25 points, then iced the game with two more free throws with 2.9 seconds left that provided the final margin.
"Once I got going, posting up and hitting 3s, I felt like I was going to have a good night, and my teammates kept getting me the ball," Singletary said. His run of consecutive points without a teammate scoring is the second longest in NCAA history, behind only the 54 in a row by Bill Mlkvy of Temple in a game against Wilkes on March 3, 1951.
Donald Sloan scored 22 points and Josh Carter added 15 for the Aggies (23-9), who punctuated a six-game winning streak by beating No. 14 Missouri on Saturday to hand the NCAA selection committee a good looking resume
-- except for the stain left by this ugly loss to the 11th-seeded Red Raiders.
"I think we were solidly in because of what we accomplished -- finishing (tied for) fourth in the Big 12
-- and Texas Tech's a pretty hot team. They did beat Kansas seven days or eight days ago. It wasn't like it was a big fluke or anything," Aggies coach Mark Turgeon said. "We'll see.
"It'll be a lot more anxious than it would have been if we would have won a game or two. I know that."
Texas Tech, which eclipsed the record of a 20-point comeback set by Kevin Durant and Texas two years ago, would have to become the first Big 12 team to win four tournament games in as many days to make it to the NCAAs.
"Normally, teams would have given up three or four weeks ago. These guys just haven't given up, and I think this is a great indication of that," Red Raiders coach Pat Knight said.
Davis opened the second half with a pair of free throws to put A&M ahead 50-29. Darko Cohadarevic then hit back-to-back baskets to get Tech's offense off to a better start than the first half, when the Red Raiders missed their first 10 shots to fall into a 15-2 hole.
But the rally really got started once Singletary -- who didn't even start
-- got rolling.