Bates, ranked No. 1 in Maine, fell to 8-1 after the best start in school history. UMF, ranked fourth in the state improved to 5-2.
The Bobcats trailed for most of the second half, but took a 61-60 lead when Rob Stockwell hit a 3-pointer with just over four minutes to play. The teams traded leads over the final minutes, with Elliott Linsley snagging an offensive rebound and laying it in to give the Bobcats a 65-64 lead with 50 seconds to play.
The Beavers took possession with just 18 seconds to play, and tough defensive pressure forced an off balance shot from the left by Teague with just five seconds to play, but the rebound caromed off the glass directly to Neilson on the opposite side of the floor.
Bates had the opportunity to pull away in the closing minutes, but went just 2-for-6 from the free throw line over the final 2:19 after making 11 of their first 14 free throws.
Travis Magnusson led all scorers with 21 points for UMF. Steve Ginn added 13 points, while Nielson had a double-double with 11 points and 10 rebounds.
Brian Gerrity scored 16 points for Bates, leaving him just seven points shy of joining the 1,000-point club. Jon Furbush also scored 16 points for Bates, while Stockwell had his fourth double-double of the season with 14 points and 10 rebounds. Linsley, who leads the New England Small College Athletic Conference in field goal percentage, scored 10 points on 5-for-8 shooting.
The Beavers used a 13-0 run over a span of nearly eight minutes to turn a 15-10 deficit into a 23-15 lead. Bates struggled from the floor, shooting just 26.3 percent in the first half to fall behind 36-29 at the break.
Bates held a 54-37 rebound advantage, and the Bobcats have yet to be outrebounded this season.
UMF hosts the University of New England at 3 p.m. on Saturday, while Bates wraps up the fall semester with another non-league home game, hosting Husson at 3:30 on Saturday.