Virgil Matthews' dunk with 4:17 left capped an 11-0 run and made it 54-44 for Montana, which has won six straight.
Weber State chipped away at the lead, narrowing it to 60-58 on Terrell Stovall's 3-pointer with 1:03 left. Montana's Kevin Criswell missed a layup before Stovall was called for an offensive foul on the other end and the Grizzlies got the ball.
Matt Martin made both his free throws after getting fouled with 14.4 seconds left to make it 62-58, but Weber State's Coric Riggs hit a 3-pointer with 4.2 seconds left to make it 62-61. Davis was fouled with 2.8 seconds and added drama by missing his first free throw before hitting his second for the final margin.
Montana's first-year coach Larry Krystkowiak screamed and raised his hands in victory as his team mobbed each other at center court.
Criswell and Martin each added 11 points for the Grizzlies (18-12), who defeated Montana State 79-67 in the semifinals on Tuesday.
Lance Allred had 21 points and 12 rebounds for Weber State (14-16), which upset top-seeded Portland State 71-61 to advance to the final.
The teams had met three times before in the Big Sky Title game 1978, 1980 and 1995 with the Wildcats winning each time. It was Montana's first appearance in the title game since 2002, when the Grizzlies beat Eastern Washington 70-66 to earn its last berth in the NCAA tournament.
Weber State beat Eastern Washington 60-57 in the tournament final the next year to earn an NCAA appearance.
Allred had just six points in the semifinal game. But the 6-foot-11 senior center had 12 points and 10 rebounds by the half as Weber State built a 28-21 lead.
Montana narrowed it to 32-28 on Matt Dlouhy's alley-oop dunk. Davis added a jumper and Martin hit a 3-pointer to briefly give the Grizzlies a 33-32 lead.
Both teams traded the lead four times until Kevin Criswell's jumper gave the Grizzlies a 47-44 lead with 6:36 to go. Davis added a pair of free throws and Martin hit another 3-pointer before Matthews' decisive dunk.
The Wildcats and Grizzlies split their two regular season games.
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