Burrage tied the game 60-60 with a jumper with 57 seconds remaining. After a Bearkat turnover, he hit the game winner with 1.9 seconds remaining with a six-foot jumper from inside the lane.
Sam Houston State (6-9, Southland 0-1), outscored UTSA 34-28 in the second period and held a 59-52 lead with 3:33 remaining after a pair of free throws by Darius Gatson.
Konner Tucker led the Bearkats with 19 points including five three-pointers.
UTSA (9-6, Southland 2-0) had two other players in double figures with Michael Hale scoring 13 and Jeromie Hill 10.
Sam Houston outscored UTSA 28-22 in the paint and picked up 24 points off the Roadrunners' 20 turnovers.
San Antonio went on an 8-0 run, including two game-clinching buckets from Kannon Burrage to steal the victory right from under Sam Houston.
One more stop at the end of the game could have resulted in overtime, but Burrage was hot and hit on seven of his eight shot attempts to lead the Roadrunners with 17 points.
Sam Houston point guard Darius Gatson's only two turnovers in the game came at the most critical point, when he was called for consecutive offensive fouls drawn by Michael Hale III. Both fouls set up Burrage's shots down the stretch.
Still, the Bearkats showed promise in different areas.
"It's always tough when that happens and they hit a last second shot with just one second to go," Tucker said after the game. "I thought we did a lot of great things today.
"We made a lot of really good plays. Defensively, we knew everything they were doing, just came up short at the end."
Sam Houston shot 43 percent from the field after struggling offensively in its non-conference schedule.
Konner Tucker led the Bearkats with 19 points and Steven Werner came off the bench to add nine points.
Even with Sam Houston's committment on the defensive end, the conference's top shooting Roadrunners went 56 percent from the field. San Antonio managed to win the game in spite of its 20 turnovers compared to Sam Houston's eight.
Bearkats head coach Jason Hooten was not happy with the San Antonio shooting 56 percent for the game and wanted to see his team win the rebounding battle, which didn't happen.
The Roadrunners' depth was critical against the Bearkats.
Hooten noted after the game that San Antonio rotated 10 players, while Sam Houston rotated eight.
However, both teams had seven players who played 22 minutes or more.
San Antonio's bench still outscored outscored Sam Houston's 33-14
"It was hard fought Southland Conference game," Hooten said. "That's kind of what we thought it would be. They've got a really good team. They're picked first for a reason. We fought back in the second half and also started off the game really well.
"But they're really deep and they kept coming at us."
The teams went back and forth in the first half with four lead changes and two tied scores.
Sam Houston's Aaron Harwell stole a pass and took it the other way to score on an up-and-under layup.
From that point, San Antonio went on a 17-7 run with consecutive 3s from Hale and Burrage, five straight points for Sei Paye, then more 3s from Igor Nujic and Stephen Franklin.
The Bearkats come away from this experience knowing they can hang with the Southland Conference tournament champions and one of the league's favorites to do it again in a game where San Antonio couldn't miss.
The Roadrunners knocked down eight-of-12 3-pointers in the first half, but missed all five long range attempts after the break.
"I think Sam is a good basketball team and had a lot to do with why we didn't play well," San Antonio head coach Brooks Thompson said after the game. "I thought in the first half we played pretty well. Then in the second half they started getting physical and really played a better second half than we did.
"The thing I'm proud of is that we made the plays down the stretch that we had to make to win."