Qualls jammed a tip-in with 0.2 seconds left on the clock to give Arkansas (12-4, 1-2 SEC) its first conference victory.
Rashad Madden also scored 18 for the Razorbacks, and Alandise Harris came off the bench to score 12 points, including six from the foul line.
James Young led Kentucky (12-4, 2-1) with 23 points. Julius Randle went 10-for-14 from the free-throw line to score a total of 20 points. Andrew Harrison scored 16, including an uncontested 3-pointer with a second left in regulation to tie the score 74-74.
Kentucky outrebounded Arkansas 50-32 but committed 17 turnovers.