While Robertson's heroics can't be minimized, a 54-yard run by Monangai gave the struggling offense a needed jolt.
The Wildcats were trailing by 13-3. Two plays after Monangai's long run brought the ball to the Tribe 21-yard line, Austin Medley scored his first touchdown of the season on a 13-yard run with 10:33 left in the third quarter.
"I think that run gave us energy, and then after that Austin Medley capped it off for us," said Monangai, who rushed for 93 yards on 14 carries, with 81 coming in the second half.
Villanova coach Andy Talley agreed.
"It was a defining moment in the game," Talley said. "It gave us a little bit of energy and spunk."
John Carpenter kicked a 24-yard field goal, his third of the game, to give the Tribe a 16-13 lead with 6:17 left.
Then Robertson went to work on a drive in which Villanova converted three third-down plays and four different receivers caught passes. Robertson rushed for 37 of his 70 yards on the final drive.
The winning drive has value for Villanova beyond this game.
"Late in the season these games will happen, maybe next week, and we will be confident we can go in the fourth quarter and pull them out," Robertson said.
After Robertson's score, William and Mary (3-2, 1-1) marched to the Villanova 40-yard line following a pass-interference call and had one final shot with two seconds left. Wildcats lineman Reggie Paris ended matters with a sack.
William and Mary used three quarterbacks. Starter Michael Graham left in the second quarter with concussion symptoms after being sacked by Rakim Cox. Redshirt freshman Steve Cluely replaced him, and then former Mainland product Brent Caprio, who had not attempted a pass while recovering from a shoulder injury, led the Tribe to a field goal in his fourth-quarter appearance.