The Bobcats (18-12) will play top-seeded LIU Brooklyn (22-8), the tournament host, in Sunday's semifinals.
With the score tied 23-23, Quinnipiac closed the half on an 18-8 run in which five different players scored. But in the second half, St. Francis rallied behind Brent Jones (21 points), Ben Mockford (17) and Stefan Perunicic (11).
The Terriers drew within a point, 49-48, on two free throws and a 3-pointer by Perunicic. But Quinnipiac went on a 9-0 run to take a 10-point lead and was never threatened.
For St. Francis, Travis Nichols scored 12 points. The Terriers (15-15) hit 21 of 57 from the field to Quinnipiac's 28 of 67.