The 32-year-old scored 25 points on Sunday to pass the milestone in his 10th season with the Cavaliers.
He now has twice as many points as Zydrunas Ilgauskas, who is second on the team's all-time list with 10,616.
James, who also played for Miami Heat, is eighth on the NBA all-time list with 27,938 career points.
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Elsewhere the Atlanta Hawks needed four periods of over-time to beat New York Knicks 142-139.
The game lasted for more than four hours, after the teams were tied after 48 minutes and then played periods of five minutes until there was a winner.
Hawk's Paul Millsap played 60 minutes, including 35 consecutive minutes, to finish with 37 points.
LeBron James became the first player to score 20,000 points for the Cleveland Cavaliers when the NBA champions beat Oklahoma Thunder 107-91.