Memphis scored 36 points in the opening period to take a 14-point lead and survived down the stretch despite making only four field goals in the final quarter.
The Los Angeles lead in the opening-round series was cut to 3-2. The Clippers will have another chance to close out the series at home Friday. A seventh game, if needed, would be played Sunday in Memphis.
San Antonio, which knocked out Utah in four games Monday, awaits the winner of the Grizzlies-Clippers confrontation.
Gasol scored 23 points and Zach Randolph came off the bench to contribute 19 for the Grizzlies. Memphis missed all six of its 3-point attempts, but held Los Angeles to 37 percent shooting and built as much as a 24-point lead late in the third period.