Harrison's career-high scoring total included a 10-for-10 day at the free-throw line and helped carry the Wildcats (13-4, 3-1 SEC) to its fifth win in six games.
Julius Randle added 18 to the Kentucky attack and Aaron Harrison scored 14.
Jarnell Stokes led Tennessee (11-6, 2-2) with 20 points and 15 rebounds. The Volunteers lost for the second time in their last three games and fell to 4-35 all-time at Rupp Arena.
The Wildcats led 34-32 at halftime but fell behind early in the second half before getting back on track.
Darius Thompson scored a 3-point jumper with 3:02 to play to pull the Vols to within four points of the lead but they could get no closer.