Louisville (13-1) has won eight in a row since suffering a five-point setback at the hands of Duke on Nov. 24.
Providence scored the game's first eight points, but Smith then came off the bench to produce 10 of Louisville's first 14 as the Cardinals battled their way back into the lead.
Louisville took a 37-31 advantage at halftime and pulled away in the second period while shooting 52 percent from the field for the contest.
Smith finished with 23 points while both Luke Hancock and Gorgui Dieng had 11. Dieng added 11 rebounds.
Providence (8-5) was paced by Kadeem Batts with 17 points and Bryce Cotton with 16.