Joe Bramanti's layup with 1:07 left gave the Wildcats (6-3) the lead for good at 57-55. Leissner extended it with a runner off the glass on the next possession and Jaleen Smith's free throws made it 61-55 with 21 seconds left.
Smith and Matt Miller added 11 points for New Hampshire and Leissner's eight rebounds also tied a career best for the freshman.
Colgate (3-8) took a 55-51 lead with a 7-0 run capped by Damon Sherman-Newsome's layup with 3:20 to play. But the Raiders missed four consecutive field goals, allowing the Wildcats to pull in front and Austin Tillotson's layup with 12 seconds left only cut the deficit back to four.
Tillotson and Sherman-Newsome scored 15 apiece.