Sykes made two free throws for the Phoenix (5-2) with 8.7 seconds left after Joe Harris missed a 3-pointer for Virginia. The Cavaliers (7-3) had a final chance to tie the score, but Justin Anderson missed a 3-point shot as the buzzer sounded.
Virginia fought back from a 13-point deficit early in the second half, tying the score at 69 on a 3-pointer from Malcolm Brogdon with 1:51 left. Brogdon led the Cavaliers with 17 points. Harris and Anthony Gill each added 12 points.
Green Bay never lost the lead down the stretch after Brogdon's score-tying 3.
Sykes, on a drive to the basket from the left side, dropped the basketball to Mays open on the right side for a lay-in to put the Phoenix back ahead at 71-69 on its next possession.
Jordan Fouse then intercepted a pass in the backcourt as Green Bay went with a full-court press on the ensuing inbound. Fouse finished off the big play with a drive down the lane for a layup to extend the lead to four points with 1:15 left. He set a single-game school record with eight steals as the Phoenix forced 14 turnovers.
A 3-pointer by Anderson at the 58-second mark cut the Virginia deficit to 73-72, leading to the frenetic final seconds of the game.
Virginia shot 50 percent (23-of-46) for the game -- three days after the Cavaliers struggled to put the ball in the basket in a 48-38 home loss to No. 8 Wisconsin. The Cavaliers were a frigid 23.4 percent from the floor in that contest.
But the 75 points by Green Bay were the most scored against Virginia this season. The Cavaliers came into the game ranked No. 2 in the country for scoring defense, allowing 52.9 points per game.
Green Bay took a 37-31 halftime lead with a seven-point run to end the half. Mays led Green Bay with 12 points in the first half, and Sykes had 11 points. The Cavaliers kept the score tight for most of the first half despite having Harris, their top scorer, on the bench with two fouls for the last 13 1/2 minutes.
After Virginia pulled within 37-35 early in the second half, Green Bay quickly pushed its lead to 13 points with an 11-0 run in less than two minutes.
Mays knocked down a 3-pointer in the fourth shot of a sequence for the Phoenix. Then after a lay-in by Alec Brown, Love hit an open 3 from the left corner in back-to-back possessions to put Green Bay ahead 48-35 with just over 16 minutes left.
The Phoenix didn't let the lead dip below eight until Harris made three straight baskets to cut the margin to 61-55 with seven minutes to play.
That was the start of a 13-3 run for Virginia, which pulled within 63-62 at the 4:01 mark.
The loss is the second straight for Virginia this week and spoiled the Green Bay homecoming for Cavaliers coach Tony Bennett. Bennett, in his fifth season at Virginia, was a star player in college at Green Bay from 1988-92 and still holds the school record for career points (2,285) and assists (601).
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Maryland point guard Melo Trimble, considered a lock to leave the program before struggling in the second half last season, has withdrawn from the NBA draft and will return for his junior season.
Louisville's Chinanu Onuaku, who recently underwent a medical procedure to correct a heart rhythm issue detected during the combine, will keep his name in the NBA draft pool.
Andrew White III will return to the Nebraska men's basketball team next season after removing his name from the NBA draft.