Tim Derksen led USF with 17 points.
The Dons led 52-47 midway through the second half when Cole Dickerson made a jump-hook, and the lead see-sawed from there for the next few minutes.
The Cougars took a 61-55 lead on a one-hander by Haws with 4:26 and the Marriott Center was rocking, but USF's Avry Holmes answered with a 3-pointer from the corner to right the Dons.
Matt Carlino's 3-pointer and Eric Mika's bucket with 2:01 remaining gave the Cougars a 67-60 lead, Derksen hit a 3-pointer from the corner to give the Dons life.
However, Collinsworth missed four free throws in the last 1:05 to give the Dons hope. But San Francisco didn't score after Derksen's 3-pointer, and fell to fourth place in the league.
The Cougars led the entire first half.
Haws was 3 of 10 in the first half, for seven points. He even missed a free throw.
The Cougars took a 21-14 lead on Collinsworth's 3-pointer with 10:28 remaining in the half, but BYU went the next six possessions with just one field goal, while USF crept back into the game.
Mark Tollefsen's driving basket got the Dons within three, 26-23, with five minutes left in the first half.
Back-to-back baskets by Holmes tied the score at 27-27, but Haws finally got going with less than two minutes left in the half and made a couple quick buckets to give the Cougars a 32-29 halftime lead.
BYU 68, San Francisco 63
• Despite shooting 14 of 29 from the free-throw line, BYU holds off San Francisco to move into sole possession of second place in the West Coast Conference
• Nate Austin grabs a career-high 16 rebounds and makes big plays at both ends of the floor in the final minutes for BYU
• Eric Mika posts a double-double with 13 points and 13 rebounds and the Cougars win the battle of the boards, 47-29