After losing the opening two games of the Western Conference quarterfinals, the Grizzlies won four in a row and will now take on Oklahoma City in the second round.
Memphis benefited from the struggles of Los Angeles star Blake Griffin, who played just 14 minutes Friday due to a sprained ankle.
The Grizzlies shot 50 percent from the field and went to the foul line 47 times, hitting 38 of those attempts.
Tony Allen added 19 points with seven rebounds and six assists for Memphis, which had lost to Los Angeles in the opening round of last year's playoffs. The Grizzlies became the 16th team in NBA history to rally from an 0-2 deficit to win a seven-game series.
Chris Paul led the Clippers with 28 points, but was ejected with 2:29 remaining after picking up his second technical foul. Matt Barnes had 30 points for Los Angeles while hitting 11-of-14 from the field.