Taquan Dean scored 19, and the 6-foot-8 Palacios shot 8-for-10 and grabbed 10 rebounds for the No. 9 Cardinals (19-3, 7-1 Conference USA), who have won eight in a row and 13 of their last 14.
Louisville led 68-64 entering the final 90 seconds of a game that was dominated by the top two field-goal percentage defenses in Conference USA.
With the shot clock ticking under 10 seconds, Louisville guard Brandon Jenkins got caught in a double team near halfcourt. Palacios came to help and when Jenkins leaped and passed him the ball, Palacios turned and hit the straight-on bank shot from five feet behind the 3-point line.
Palacios had made only 16 3-point shots all season coming into the game.
Louisville went 4-of-4 from the free-throw line in the final minute to keep No. 18 Cincinnati at bay.
Eric Hicks matched a career high with 21 points to lead Cincinnati (17-4, 6-2), which lost to Louisville for the sixth time in the last 10 meetings between the rivals.
The Bearcats missed 13 of their first 18 shots in the opening 10 minutes, but trailed only 15-12 because of seven Louisville turnovers.
Louisville led 25-24 when a 3-pointer by Dean started a 7-0 spurt.
Palacios' steal and a fast-break layup in the first minute of the second half gave Louisville its biggest lead at 40-31.
James White then scored in the lane to trigger a 9-0 Bearcats' run.
The Cardinals rebuilt a four-point lead, but Francisco Garcia, Louisville's leading scorer, got his fourth foul and left the game with 9:48 remaining. The Bearcats couldn't take advantage, though, going four minutes without a field goal.
Without Garcia, Dean became Louisville's go-to guy and sank two 3-pointers to push the lead to 64-55. Garcia returned with 3:59 left and the Cardinals ahead 66-60, but he fouled out a minute later, clipping Armein Kirkland's arm on a layup. Kirkland hit two free throws to trim Cincinnati's deficit to 67-64.
A free throw by Jenkins with 2:08 remaining put the Cardinals up 68-64.
The Bearcats held Garcia to 11 points on 5-of-14 shooting, but they shot poorly themselves, going 23-of-61 from the field (38 percent) and 4-of-18 from 3-point range (22 percent).