After Luc Richard Mbah a Moute made the second of two free throws, James scored on a layup. Redick made a jumper, but Shane Battier and Allen closed out the Bucks with a pair of 3s, with James credited with the assist on each.
He then converted a three-point play and added a layup to finish up the run.
"At some point during that stretch right there, (James) decided he was going to put his imprint on the game and he did. In a big way," Bucks coach Jim Boylan said. "When you're a superstar player like he is, that's what superstar players do."
Superstar teams, too.
Though Milwaukee had chances to win Games 2 and 4 late, and led Game 3 by 10 in the first half, there was never a sense Miami would lose any of them. Any time the Bucks showed the slightest bit of momentum, the Heat were quick to snuff it out.
"They'd either make a big play or it was a turnover on our part," Boylan said. "When you're playing against a quality team like Miami, you make some mistakes, they usually capitalize on them. And they did that."
The Bucks had seven turnovers before the first quarter ended Sunday, and shot just 37 percent for the game. Ellis led the Bucks with 21 points, and Larry Sanders had 11 rebounds to go with seven points.
But Milwaukee got almost nothing again from Brandon Jennings, who was benched for the entire fourth quarter. Jennings, who had guaranteed the Bucks would win the series in six games, finished with three points on 1-of-7 shooting.
After scoring 26 points in Game 1, Jennings had 27 total in the final three.
"We've been playing close, playing hard," Ellis said. "They're just a better team."
And that is what makes them so dangerous at this time of year.
Despite their star-studded roster, the Heat have little use for individual accolades or accomplishments right now. They have one goal in mind, and one goal only, and it will take the entire team to achieve it.
"I'm glad to get this," coach Erik Spoelstra said. "It's the first step in our journey. Hopefully, a long journey."
NOTES: This is the first time the Heat have swept a playoff series since James and Chris Bosh came to Miami three years ago. ... The Heat have won eight straight postseason games dating to last season. That matches the franchise record. ... NBA Commissioner David Stern was in attendance. ... James scored 30 or more for the 54th time in the postseason, second only to Kobe Bryant among active players. ... The Bucks are 20-26 in elimination games. ... Packers LB Clay Matthews was at the game.