Phoenix led 90-80 with seven and half minutes left but the Heat staged a remarkable rally to turn the game on its head and give Miami their 14th straight win at American Airlines Arena.
The Suns failed to score a point for more than six minutes and the Heat took full advantage, making nine free throws during the run and grabbing a 97-90 lead once the smoke had cleared.
"It was either now or never," said Miami's Dwyane Wade, who had 19 points. "They were playing very well. We were down 10 and had to make a move."
Chris Bosh scored a game-high 29 points, while LeBron James added 20 points, eight rebounds and six assists to help Miami (34-11) pull within 2.5 games of conference-leading Chicago.
Phoenix (23-23), who had ridden a four-game winning streak into playoff contention, got a team-high 19 points from Grant Hill.
Hill and James collided while chasing the ball in the final seconds of the game and Miami's two-times MVP remained on the floor for a couple minutes before returning to his feet.
The Heat played without starter Joel Anthony, who missed the game with a sprained ankle and was replaced by Dexter Pittman.
The Suns' Marcin Gortat scored 13 points and Steve Nash had nine and 10 assists.
"We want to make sure this is the toughest place to play in the NBA," Bosh said. "The playoffs are approaching so we have to keep that mentality - defend our home court."
