James' driving dunk and subsequent foul shot put Miami up 84-77 with 3:47 left and the Heat went on to claim the victory, giving them a 2-games-to-1 advantage in the championship series.
James also grabbed 14 rebounds. Dwyane Wade added 25 points, seven rebounds and seven assists in the victory.
Chris Bosh chipped in 10 points and 11 rebounds as Miami cleaned up at the free-throw line, going 31-of-35.
Kevin Durant led Oklahoma City with 25 points and Russell Westbrook contributed 19 for the Thunder, who could manage only 4-for-18 from 3-point range.
The Thunder will try to even the series in Tuesday's Game 4 at Miami.
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