Napier and Omar Calhoun hit 3-pointers in overtime to give the Huskies (14-5, 4-3 Big East) a 79-76 edge with 42 seconds left. Napier hit 3 of 4 free throws in the final 30 seconds to secure the victory.
DeAndre Daniels also scored 18 points for UConn, whose 50 percent shooting (27 of 54), including 52.9 percent (9 of 17) outside the 3-point line, proved too much for Providence (10-11, 2-7), which shot 33.8 percent (22 of 65).
The Friars grabbed 55 total rebounds to the Huskies' 24.
Neither team led by more than three points during the second half, when Boatright scored 12 of his points. Bryce Cotton's 3-pointer with 1:30 left pulled Providence into a 66-66 tie, and LaDontae Henton's layup with 18 seconds remaining for the Friars sent the game to overtime.
Kadeem Batts scored 20 points and Cotton 18 for Providence, and Henton had 15 points and 13 rebounds.
UConn took an early 12-3 lead, as Providence didn't get its first field goal for 5 minutes. A 9-1 run put the Huskies ahead 25-10, but Providence scored the next eight points and closed the half with a 12-3 run to trail 33-32.