Ryan Boatright added 20 points and Omar Calhoun had 10 points as the Huskies (16-6, 6-4 Big East) rallied from a seven-point deficit and handed Seton Hall (13-11, 2-9) its fifth straight loss and ninth in 10 games.
Napier scored 18 of his 22 points in the second half and finished 8 of 12 from the field while playing 39 minutes. He hit a free throw, a fast-break layup and a 3-pointer in a 9-0 run that gave Connecticut the lead for good.
The Huskies got sloppy in the final eight minutes of the first half and blew a 13-point lead.
Seton Hall led by one at the half and pushed the margin to 43-36 early in the second half as Cosby and Edwin hit 3-pointers to ignite an 8-0 spurt.
After Seton Hall pushed the lead back to 46-42, Napier, who didn't make his first basket until late in the first half, took over. He scored six in the nine-point run with Niels Giffey adding a three-point play.
Seton Hall stayed within striking distance the rest of the way, but Boatright put the game out of reach with a long 3-pointer and a dunk off a fast-break setup by Napier, who had a steal with about four minutes to play.
Playing with Brandon Mobley and Brian Oliver sidelined, Seton Hall didn't seem to have the talent to stay with the Huskies early. Connecticut used an early 11-0 run that featured a four-point play by Boatwright in taking a 13-5 lead, and it upped the margin to 25-12 with a 7-0 run that included four points by DeAndre Daniels.
Seton Hall suddenly picked up its defense and got 10 straight points from Edwin in a 16-0 run that gave the Pirates a 28-25 lead.
Connecticut got within a point three times in the half but didn't regain the lead until Napier scored in the lane on the first possession of the second half.