The Grizzlies were within eight points, 53-45, with 12:08 left after a twisting, off-balance layup by Kareem Jamar.
But the Cougars (5-2) went on a 14-5 run to seal their third straight victory this season and sixth straight at EnergySolutions Arena, home of the Utah Jazz.
Haws started the run with a pair of free throws, and Craig Cusick hit his second 3-pointer of the night to ruin any hopes the Grizzlies had of extending their three-game winning streak.
Haws was trying to tie the school record of scoring at least 20 points in the first seven games of the season, but came up two points short.
“This was a great team win," Davies said. "Both halves, we played solid, and the second half we came out and played even stronger. I thought we shared the ball pretty well. Everyone was finishing plays and playing on the attack. When we play like that, we are hard to be beat.”
“I thought we played well," said head coach Dave Rose. "I was really pleased with the start of the game. Defensively we got out to a great start and challenged every shot. Offensively, we had open shots, we hit them and got off to a nice lead. We probably could have shot free throws a bit better, but other than that I thought it was a pretty good game.”
Jamar led Montana (3-2) with 17 points and nine rebounds and Keron DeShields added 12 points.
BYU next travels to Iowa State to face the Cyclones Saturday afternoon.