Playing their second game on back-to-back nights, the Lakers fell behind by as many as 22 points before rallying to within three, 106-103, with 14.9 seconds left on a long Jordan Farmar three-point shot.
Jodie Meeks was the game's leading scorer with 19 points on 7-of-10 shooting. Six Lakers hit double figures, including Johnson (18), Pau Gasol (17), Farmar (16), MarShon Brooks (14) and Kent Bazemore (14).
Memphis shot 48.9% from the field and finished with an 11-rebound advantage, 45-34.
Lee led the Grizzlies with 18 points. Zach Randolph needed 14 shot attempts to score 13 points.
Both Gasol brothers notched double-doubles. Marc Gasol scored 14 points with 12 rebounds and five assists. Pau Gasol had 10 rebounds and three assists.
The Lakers shot 49.4% from the field and 53.6% from three-point range (15 of 28). Johnson and Farmar both hit four apiece.
Johnson also blocked four shots, but the Lakers' defense was otherwise porous in the first half, giving up 65 points. Memphis scored a more reasonable 43 points in the second half while the Lakers rallied for 57.
The Grizzlies (32-24) will next visit the Oklahoma City Thunder (43-15), also on Friday.
The Lakers rallied in the third quarter, producing a 17-4 run to get back into the game in Memphis after a poor first half.
Jodie Meeks has scored 19 points on 7-of-10 shooting while MarShon Brooks has 14 off the bench without missing a shot.
The Grizzlies are shooting 51.4% from the field. Marc Gasol has notched a double-double with 13 points and 11 rebounds.
The Lakers improved their shooting to 49.1% from the field but have struggled handling the ball, committing 14 turnovers. The Grizzlies had six third-quarter turnovers and 10 for the game.
The Grizzlies had an easy half against the Lakers, taking a 19-point lead into the break.