Lowry added 11 rebounds and 12 assists.
Luther Head hit a 3-pointer and Carl Landry had a three-point play to give the Rockets a 107-101 lead with 2:17 to play in the second overtime.
But Memphis came back to tie it on consecutive baskets by Rudy Gay, Lowry and Casey Jacobsen with 43 seconds to play. Lowry had eight points in the two overtimes.
Jacobsen finished with 17 points for the Grizzlies.
Stromile Swift sparked a second-half rally to help the Grizzlies come back from a 12-point deficit in the first half. Swift had 16 points, including 13 in the third quarter.
Trailing by one to start the fourth, Memphis got seven points from Juan Carlos Navarro to pull even at 70 with 7:53 to play.
Houston's Steve Francis scored 13 points, six in the first overtime.
Francis, who signed as a free agent in the offseason with his former team, scored five points in the fourth quarter to help Houston come back from a 79-75 deficit.
Rockets starting guard Rafer Alston sat out the game with a bruised right thigh and is day-to-day. Mike James started in his place and led Houston with 19 points.
James had 11 points, making all three 3-pointers, in the first quarter. The Rockets led by as many as seven during the period and were up 21-16 at the end of it.
Bonzi Wells came off the bench in the second quarter and added six points as Houston led by as many as 12 before settling for a 42-33 halftime lead.