After Jodie Meeks hit a three-pointer with the Lakers down, 85-81, with 1:20 left, Randolph was able to seal the game with 16.4 seconds left, hitting a driving finger roll around Jordan Hill to put the Grizzlies up three.
Steve Blake missed a three-point attempt to tie and Randolph hit a pair of free throws to finish with a game-high 28 points.
The Lakers had one last gasp when Nick Young was fouled on a three-point attempt with 6.5 seconds left. Young hit the first two from the line, but missed the third. The Grizzlies were able to grab the rebound and run out the clock.
Jodie Meeks led the Lakers with 25 points on 10-of-16 shooting. The team shot 43.8% from the field and 41.7% from three-point range.
Young scored 18 off the bench. Pau Gasol notched a double-double with 10 points and 13 rebounds.
Randolph also had a double-double with 11 boards. Marc Gasol neared a triple-double with 18 points, eight rebounds and eight assists. The Grizzlies shot 43.5% from the field but just 7.7% from three-point range (1-13).
The Lakers left five points at the free throw line (6-11) while Memphis hit all 14 of their attempts.
Memphis (4-5) will travel to Sacramento to play the Kings, also on Sunday.
Lakers 66, Grizzlies 63 (end of third quarter)
The Lakers pulled ahead in the third quarter by a small margin, led by a hot shooting performance from Jodie Meeks.
Meeks led all scorers with 22 points in 29 minutes while hitting six of nine shots. The Lakers as a team hit 48.3%.
The Grizzlies were led by Zach Randolph with 14 points, shooting 40.3% collectively.
Marc Gasol neared a triple-double with 10 points, eight assists and five rebounds.
Lakers 40, Grizzlies 40 (halftime)
The Lakers rallied back from a 10-point deficit to tie the Grizzlies by the half. Nick Young scored the final six points of the quarter, which included a basket while being covered by noted defender Tony Allen.
Young led the Lakers with 12 points on 5-of-8 shooting. The Lakers hit 40.5% from the field in the half.
Zach Randolph had 12 points for Memphis, which shot just 35.6% from the field.
Jodie Meeks provided a spark for 10 points while Pau Gasol nearly notched a double-double with eight points and 11 rebounds in 17 minutes.
Grizzlies 24, Lakers 17 (end of first quarter)
Both the Lakers and Grizzlies struggled to hit shots in the first quarter.
The Grizzlies shot 38.1% from the field, led by Zach Randolph's 10 points.
The Lakers missed 16 of 23 shots, converting just 30.4%. While the Grizzlies converted all eight of their free-throw attempts, the Lakers only made two of five.
Pau Gasol led the Lakers with six points.
The Lakers (4-6) host the Memphis Grizzlies (3-5) on Friday night at Staples Center.
The Grizzlies came into the season with high expectations but they've struggled under rookie Coach David Joerger.
For a more in-depth breakdown, check out Preview: Lakers vs. Grizzlies.