The victory was the first for the Big Red under coach Bill Courtney, who replaced Steve Donahue this season. Donahue left to coach Boston College after leading Cornell to a 29-5 season, an Ivy League record for victories, and a trip to the NCAA Sweet 16.
With the score tied at 52, Groebe hit a 3-pointer to put Cornell in front to stay, 55-52, with eight minutes remaining. Osgood's bucket with three minutes left gave the Big Red its largest lead of the game, 64-56.
Albany, of the Metro Atlantic Athletic Conference, which led by 12 in the first half, scored the next five points and had a chance to tie, but Tim Ambrose's driving shot in the lane in the final seconds missed.
Ambrose led all scorers with 20 points.