An All-Star for the first time this season, Brand had a double-double by halftime with 16 points and 13 rebounds. His dominance inside allowed the Clippers to take a 46-29 lead after 24 minutes.
But Los Angeles nearly let the game slip away in the third quarter as Memphis went on a 21-7 run over the first 6:52.
In the third quarter, Memphis shot 64 percent (14-of-22) and trailed just 68-62 after 36 minutes.
But the Grizzlies went cold from the field, making just 1-of-11 shots to open the quarter. While Memphis was struggling, Los Angeles' Jeff McInnis took over, scoring nine of his 19 points.
The Grizzlies dressed just eight players and were led by Eddie Gill, who had 20 points. But Gill, who signed a 10-day contract Saturday, had just five in the fourth quarter.
With the win, the Clippers (28-30) moved three games behind idle Seattle for the eighth and final playoff spot in the Western Conference.