LeBron James scored 24 points and Dwyane Wade scored 22 points with 10 rebounds to overtake the Mavericks after trailing by eight points in the third quarter. Chris Bosh had 10 points and Mario Chalmers had 12.
The Mavericks' Dirk Nowitzki scored 27 points and took eight rebounds. Shawn Marion scored 16 points and Jason Terry notched 12 for Dallas.
Miami has won five straight games against Dallas in finals games going back to 2006.
Game 2 is scheduled for Thursday at 6 p.m. PDT on ABC7.
The Associated Press contributed to this report.