Titus Rubles, a junior-college transfer making his Bearcats debut, came off the bench to score eight points and grab seven rebounds.
The game drew 5,728. A couple of miles south, the Cincinnati Bengals hosted the New York Giants and beat the Super Bowl champions 31-13.
Cincinnati opens the season with four straight home games against unranked opponents, with Mississippi Valley State visiting on Tuesday. Last season, the Bearcats reached their first-ever Big East tournament championship game and made it to the NCAA's round of 16 for the first time since 2001.
Cincinnati improved to 91-21 in season openers, including 80-10 at home. Cincinnati has won 47 of its last 48 season openers at home, the only loss coming to Belmont 86-75 in the 2007-2008 tip-off.
The win was the 114th for Bearcats' coach Mick Cronin, snapping a tie with Ed Jucker for fifth place on the program's career coaching wins list. Cronin, starting his seventh year at his alma mater, is 114-88. Jucker guided Cincinnati to NCAA championships in 1961 and 1962.
The Bearcats are honoring All-American and longtime NBA star Jack Twyman by wearing a patch on their uniforms with his No. 27. Twyman, who played in the mid-1950s, died in May.
Washington scored the game's first two points, giving Tennessee-Martin its only lead 1:04 into the game before Cincinnati took control with a 14-0 run capped by Ge'Lawn Guyn's 3-pointer with 13 minutes left in the first half.
The Bearcats later took advantage of a Skyhawks' 6:36 scoring drought to run off 17 unanswered points, building their largest lead of the half at 33-6.