Despite the efforts of the four-time MVP, Miami struggled to pull away from a determined Nets side, who were looking to level the series in front of their own supporters.
James, Paul Pierce, Shane Battier and Alan Anderson were all given technical fouls in a fiercely competitive opening quarter, but it was the visitors who started the better as they edged to a 27-22 lead after 12 minutes.
Despite being unable to prevent the defending champions from storming seven points clear at the half-time interval, Brooklyn were able to outscore their rivals 27-23 in the third quarter as Joe Johnson top-scored for the hosts with 18 points.
The two sides were level heading into the final minute of the game, but Chris Bosh's three-point effort put the Heat ahead for the final time in the contest.
Ray Allen then recorded four free throws, before a successful attempt from the line gave James his 49th point and secured the win for Miami.
Erik Spoelstra's side can book their spot in the Eastern Conference finals by beating Brooklyn in game five on Wednesday.