The Bearcats trailed for much of the contest, but rallied down the stretch to remain unbeaten in the American Athletic Conference.
Cincinnati had a two-point lead with 48 seconds remaining when Rubles was fouled. He missed badly on both of his attempts from the line, giving Connecticut an opportunity to pull off the upset.
Shabazz Napier then missed a 3-point try for the Huskies and Rubles grabbed the rebound. Before Rubles could pass the ball he was fouled again, but this time the 6-foot-7 forward converted both of his tries from the line to give the Bearcats a four-point cushion.
Sean Kilpatrick led Cincinnati with 25 points and 12 rebounds. The Bearcats improved to 22-2 and are 11-0 in the AAC. Defending NCAA champion Louisville is second in the conference race at 8-2.
Connecticut (17-5, 5-4 AAC) was paced by Napier with 16 points.