Allen finished with 19 points, Mario Chalmers and Andersen each scored 12 and Bosh finished with 11 for the Heat, who played without Dwyane Wade.
One stop was all Miami needed from there, and the Heat found a way.
"They're an awesome offensive team," Bosh said. "They've been on a tear lately. We see why. ... It was difficult all night, but eventually we found a way."
It was all part of a game that had wild ebbs and flows.
The Hawks were scoreless for the game's first 4:48, falling behind big as the Heat had everything working early — even a reverse dunk in transition from the suddenly spry Allen, who has been fighting tendinitis in his right knee for a few days. Shane Battier dove on the floor for a steal and to knock the ball ahead to James, who got loose for a dunk that put Miami up 13-0.
It all changed in a hurry in the second.
Atlanta shot 72 percent; the Heat shot 32 percent. The Hawks outscored the Heat 31-17 and had a 54-50 lead at the break, fueled largely by Korver taking four shots, all from 3-point range, all of them good.
Millsap went 4 for 4 from beyond the arc in the third, Atlanta made six in all and led by as many as 11 before going into the fourth up 84-77.
"There were a lot of good things we did," Millsap said.
NOTES: Korver's NBA-record streak of consecutive games with a 3-pointer is now at 97. ... The Heat ran a tribute to Nelson Mandela on the arena screens at halftime, narrated by Alonzo Mourning, who spoke of his time meeting the former South African president who died earlier this month. ... Including playoffs, the Heat went 47-7 at AmericanAirlines Arena in 2013. ... James has scored in double figures in 522 consecutive regular-season games, four shy of matching Moses Malone for the NBA's fourth-longest such streak.