With All-Star game MVP Dwyane Wade missing his second consecutive game with a strained left calf, Kidd knocked down a 3-pointer with 4:21 left that gave Dallas the lead for good.
Kidd sees the season winding down and said it is time for the Mavericks, who have the fourth-best record in the West, to make a push.
"We had 30 games left after the All-Star break," said Kidd, who is second all-time in assists with 10,705. "Being a veteran ballclub, we understand what's at stake. So the dress rehearsal is over and we've got to get serious."
Wade, who was the NBA Finals MVP in 2006 when the Heat beat the Mavericks, was in a suit and a protective walking boot as he watched his team's five-game winning streak end. Without their star on Friday, the Heat beat the Memphis Grizzlies 100-87 in double overtime.
On Saturday, Miami fizzled offensively down the stretch. The Heat scored 12 points in the third quarter and couldn't match the Mavericks' offensive surge late.
"It's a lesson for us that regardless of whether Dwyane plays or not, our formula and game plan have to be the same," Heat coach Erik Spoelstra said. "Collectively, we can make each other better. But if the ball stops moving, it becomes a stagnant game for us."
Kidd scored 19 points in the second half as the Mavericks rallied from an eight-point halftime deficit to beat Miami for the 12th straight time in the regular season.
Caron Butler had 20 points and Kidd had 11 assists for the Mavericks, who have won three in a row. Kidd also had three steals and is one shy of tying Scottie Pippen for fifth place on the career list.
Kidd hit all three of his shots and six of seven free throws in the fourth.
"After the All-Star break, he's been phenomenal," Nowitzki said. "He's been pushing the ball and scoring."
Daequan Cook scored a season-high 22 points, and Jermaine O'Neal added 18 points and 13 rebounds for the Heat.
"I think overall we played well without Dwyane," Miami forward Udonis Haslem said. "We gave ourselves a chance to win. They just made plays and we didn't."
Kidd hit a 3-pointer that gave Dallas a 71-64 lead after three and then scored his team's first 11 points of the fourth. He was one point away from tying his season-high point total.
Miami's Mario Chalmers drove for a layup to make it 79-79 with 4:34 left.
Kidd followed with a 3-pointer, Nowitzki hit two free throws and Brendan Haywood put back a miss to make it 86-79 with 2:57 remaining.
The Mavericks hit 9 of 10 free throws in the final minute as they increased their lead in the Southwest Division to 2½ games over the San Antonio Spurs.
"We had to fight and scrap to get the win," Haywood said. "We didn't have the energy we needed in the first half. Dirk and (Kidd) had a great game and they really pulled us through."
Miami jumped out to a 52-44 halftime lead before Dallas rallied with a 19-5 run over the final 8:07 of the third.
Shawn Marion, who scored six points in the spurt, drove for a layup to tie it at 60. Nowitzki hit a jumper to give Dallas a 66-64 lead, and Kidd connected on a 3-pointer to stretch the lead to seven with 1:53 remaining in the third.
The Heat went 5 of 13 from the floor in the third quarter and committed six turnovers.
NOTES: Mavericks center Erick Dampier (right middle finger surgery) missed his third straight game. ... The Heat are 40-60 when Wade has missed a game in his career. ... DeShawn Stevenson, who was acquired from Washington, along with Butler and Haywood, saw his first action with the Mavericks, playing the final 25 seconds of the first half. ... Kidd banked in a shot from halfcourt, but he released it just after the halftime buzzer sounded.