Dyer scored 16 points in the second half, helping the Cougars (14-9, 5-4 Conference USA) get their 1,000th win in school history. Houston improved to 1,000-672 in 60 years of basketball.
The Cougars are now 10-1 at home this season, with its only loss coming against Tulane on Jan. 26.
Memphis, which ranks 13th out of 14 C-USA teams in assist-to-turnover ratio, held a 30-25 halftime lead but was ultimately doomed by tying a season high with 26 turnovers.
The Cougars ran off 14 straight points to take a 39-32 lead early in the second half, holding Memphis scoreless for almost seven minutes. During that stretch, the Tigers committed seven turnovers and missed seven shots.
Rice hit a 3 to end the Tigers' drought with 11:25 to go, and Memphis quickly tied the score up at 39.
But Houston went on another late run, with Dyer hitting a 3 and Latham stealing the ball from Washington and scoring on a layup to give the Cougars a 52-44 lead.
Memphis would never get any closer down the stretch.
The Cougars won despite getting just eight points from leading scorer Andre Owens, who averages 18.1 points a game.