Dwyane Wade played through a nagging knee injury to finish with 18 points and six assists and Chris Bosh had 12 points and seven rebounds for the Heat, who won four straight games against the undermanned Bulls after dropping the opener in Miami.
"We knew right from the very beginning of this series that we would have to earn everything and that proved to be true," said Miami coach Erik Spoelstra.
The defending champions Heat will face either the Indiana Pacers or the New York Knicks in the semis. The Pacers lead that quarter-final series 3-1 with Game 5 set for Thursday at Madison Square Garden.
James also had seven rebounds for the Heat who reached their third straight Eastern Conference final series.
The injury-plagued Bulls had two chances in the final seconds to tie it, but Nate Robinson and Jimmy Butler both missed from beyond the arc.
Carlos Boozer finished with a game high 26 points and 14 rebounds while Robinson had 21 points and Butler 19 for the Bulls, who played the entire series without Derrick Rose, Luol Deng and Kirk Hinrich.
The Bulls have been decimated by injuries. Derrick Rose has not played since he tore his ACL in the first play-off game last season.
Tom Thibodeau, the Bulls coach, blamed the loss on the injuries.
"Right now health," he said when asked what the difference in the series was. "That's the biggest thing. But they are a great team and you have to give them credit.
"They are not going to beat themselves. You have to beat them. You have to play well for 48 minutes."
Thibodeau said he had no regrets about not trying to bring Rose back for the series.
"If we were going to make a mistake we wanted to make the mistake on the side of caution," he said.
Miami's Wade has been plagued by knee troubles of late and only managed six points in game four. He rebounded Wednesday going seven-of-13 from the field and a perfect four-of-four from the foul line in the win.
"He finds a way to make big time plays. It's great to see him respond like that," Spoelstra said.
Miami appeared poised for a smooth closeout victory by getting off to a 22-4 start to Wednesday's contest.
Chicago chipped away at the Heat lead, and the Bulls led by six at halftime. They remained in front for the third quarter and led 77-69 going into the fourth.
Shane Battier nailed two from beyond the arc in Miami's 13-4 run to open the final quarter. Norris Cole hit a jump shot from the left side for an 82-81 Heat lead.
Wade drained consecutive jumpers to keep the Heat in front then had a slam dunk off a miss to put the Heat ahead 93-86 with three minutes left.
Memphis Grizzlies 88, Oklahoma Thunder 84
Zach Randolph helped Memphis reach the Western Conference finals for the first time in franchise history, finishing with 28 points and 14 rebounds in the Grizzlies' Game 5 victory over Oklahoma City.
Oklahoma City's Kevin Durant missed a 16-foot jumper from the left wing that would have tied it with 6 seconds left, finishing off a miserable shooting night for the three-time NBA scoring champion.
Durant ended up with 21 points on 5-for-21 shooting, the third-worst performance of his play-off career. The top seed in the West, Oklahoma City went 2-6 after All-Star Russell Westbrook went out with a knee injury.
Mike Conley added 13 points and 11 assists for Memphis.
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