McCleskey rushed for 101 yards and three scores to establish a conference mark with 34 career touchdowns. He surpassed Ibn Green's previous standard of 32.
Last week, McCleskey set a Conference USA record for career rushing touchdowns with 30.
He tied Green's mark with a three-yard touchdown run less than three minutes into the contest that staked the Bearcats (3-5, 2-2) to a 7-0 lead.
McCleskey established the record when he scored from a yard out on the final play of the firsty quarter to make it 14-3. His record-setting touchdown was set up when Memphis running back DeAngelo Williams fumbled at the five.
McCleskey scored his final touchdown with 3:49 remaining in the first half, when he ran in from four yards out to extend Cincinnati's lead to 21-3.
McCleskey also moved past Robert Copper for fourth place on the school's all-time rushing yards list with 2,873.
Cooper had 2,786 from 1996-99 and McCleskey has five games to join Reggie Taylor as the only 3,000-yard rusher in school history.
Cincinnati's Gino Gudugli completed 21 of 34 passes for 225 yards while Memphis' Danny Wimprine was 13 of 27 for 154 yards with three interceptions.
The Bearcats had 217 yards on the ground and 438 total yards of offense. Tigers (2-6, 1-4) managed just 47 rushing yards and 220 overall.