Andrija Matic scored eight points for the Eagles (5-15, 3-6 Patriot), who shot 8 of 18 from long distance and 14 of 21 from the free-throw line (67 percent).
Trailing by one at halftime, Colgate's Jordan Swopshire hit a 3-pointer and the Raiders led 27-25 early in the second half.
The lead flip-flopped until the Eagles scored eight unanswered points for a 42-36 edge, then pulled ahead 47-38 on Charlie Jones' 3-pointer with 6:32 left.
Colgate closed to within 49-47 on Alex Ramon's 3-pointer with 1:15 to go, but Reed replied with a 3 and Jones added two free throws down the stretch.
Tom Rivard scored 15 points and Austin Tillotson 10 for Colgate (10-11, 6-4), who have lost two straight.