Villanova led 20-7 at the half but had four punts in the second half before Randall Harris and Evan Horn ended the last two possessions with interceptions. Horn's second pick, inside the 10, came with 14 seconds left.
Brosmer completed 25 of 32 passes for 276 yards and one interception. He capped an 80-yard game-opening drive with a 15-yard strike to Brian Espanet. Villanova scored three touchdowns, with Daniel Smith running for one and passing for another, to build the halftime lead.
The first Brosmer-Laube scoring hookup, covering 21 yards, put New Hampshire up 21-20 with 5:34 to go in the fourth quarter. The second, for 21 yards, came with 8:49 to play.
New Hampshire (5-3, 4-1 Colonial Athletic Association), had 403 yards of total offense.
Villanova (6-3, 3-3) piled up 398 yards with Smith going 21 of 40 for 313.
Georgia coach Kirby Smart: 12-team playoff would be college football's 'greatest change'
Georgia's Kirby Smart, LSU's Ed Orgeron and Coastal Carolina's Jamey Chadwell all shared their thoughts on the possibilities of an expanded College Football Playoff.
Ex-Colorado QB Sam Noyer heads to Oregon State for final season
Former Colorado quarterback Sam Noyer will use his final year of eligibility at Oregon State.
College Football Playoff expansion: 12-team historic simulations, storylines and more
Bill Connelly crunches the numbers using SP+ to figure out the most likely scenarios for how a 12-team playoff would have played out since 2014.
Latest on Bo Schembechler ignoring warnings of sexual misconduct from doctor
Michele Steele reports on claims that the late Bo Schembechler ignored warnings about former team doctor Robert Anderson's sexual assaults.
Herbstreit calls CFP expanding to 12 inevitable and for the best
Kirk Herbstreit joins SportsCenter to discuss the College Football Playoff possibly expanding to 12 teams.
Former Michigan football players say coach Bo Schembechler ignored warnings, son's complaint about doctor's abuse
Former Michigan football players Dan Kwiatkowski and Gilvanni Johnson, who have told investigators they were molested by former team doctor Robert Anderson, said head coach Bo Schembechler ignored their warnings at the time about abusive treatments.