With the win, the Heat improved to 9-0 in the "Big Three" era of James, Wade and Bosh in home games in which they had the opportunity to close out a playoff series.
In addition to Johnson's contributions, the Nets got 19 points from Pierce and 17 from point guard Deron Williams.
Johnson had 12 points in the third quarter as the Nets outscored Miami 26-24 and padded the seven-point lead they held at halftime. The Nets went into the fourth ahead 75-66.
There were 13 lead changes in the first half, but the Nets were up 49-42 at the break.
Cold-shooting Miami shot 34.1 percent from the field in the first half, including one of 16 on 3-point attempts.
After the game, Nets first-year coach Jason Kidd seemed encouraged by the big picture.
"It's a process, and we just put this together," Kidd said of how quickly the team was assembled during the last offseason. "To get to the second round gives us something to build on."
Meanwhile, the Heat is headed to the Eastern Conference finals for the fourth consecutive year.
"It's an incredible accomplishment," Allen said. "It feels great, but the job is not done yet."
NOTES: It is possible that Wednesday's game marked the end of the 19-year NBA career of Nets C Kevin Garnett, who turns 38 on Monday. Garnett averaged a career-low 6.5 points in the regular season. In the opener of the Heat series, he was held scoreless in a playoff game for the first time in his career, and he scored just two points in Game 5. ... The future of Nets F Paul Pierce, 36, also is uncertain. He will become an unrestricted free agent on July 1. ... In a statistical oddity, reserve G Ray Allen led Miami in rebounding twice in the first four games of this series. F LeBron James grabbed a game-high nine rebounds Wednesday. ... Heat G Dwyane Wade, who was rested repeatedly in the regular season, is healthy now. He has yet to miss a postseason game and is averaging 34 minutes. ... Miami made 29 of 31 foul shots Wednesday. Brooklyn had just 15 attempts, making 12.