James forced overtime with a layup before scoring late in the extra period as the Lakers extended their lead to 3-1 in the Western Conference play-offs.
The Lakers can seal victory and eliminate the Grizzlies in game five on Wednesday.
It is the first time a Lakers player has reached 20 or more points and rebounds in a single play-off game since Shaquille O’Neal in 2004.
D’Angelo Russell also scored 17 points for the Lakers, including three long-range three-pointers in the fourth quarter.
Earlier on Monday, Jimmy Butler scored an impressive 56 points as Miami Heat beat the Milwaukee Bucks 119-114 in the Eastern Conference play-offs.
Butler’s haul is the joint fourth highest individual points tally in play-off history.
Speaking after the game, Butler credited his performance to his team-mates.
“This legit was a complete team effort,” said the six-time NBA all-star.
“I know everybody sees the 56 points, but if my team-mates aren’t looking to get me the ball, or setting great screens where I can get to my right or my left or the free throw line, this game’s a lot different.
“A lot of shots went in tonight. But my team-mates kept feeding me the ball and telling me to attack. When you’ve got team-mates like that, good things happen.”
Victory for the Heat stretches their series lead over the top seeded team to 3-1.