Will Kelly scored the only field goal for Navy (12-5, 3-1 Patriot League) in overtime as the Midshipmen shot 11 for 14 from the free-throw line in the extra session.
Navy took a 40-31 lead on Tim Abruzzo's layup with 12:41 left in regulation. But Austin Tillotson sparked an 18-4 Raiders run in which he scored half the points. Abruzzo's layup with 24 seconds left tied the game at 58.
Tillotson finished with 21 points while Tom Rivard and O'Brien both added 13 for Colgate (6-9, 2-2).
Navy has won three of its first four conference games for the second-straight season.