ATLANTA -- Trae Young got his last shot to bounce in, Jimmy Butler missed his final heave and the Hawks beat the Miami Heat 111-110 to draw within 2-1 in their first-round playoff series with the East's top-seeded team.

"I didn't have any doubt that if I shot it, I wasn't going to make it," said Young, whose floater with 4.4 seconds left capped a 16-point second-half comeback by Atlanta.

The eighth-seeded Hawks, who needed two play-in tournament wins just to earn a berth in the playoffs, seemed headed for a 3-0 series deficit at several junctures on Friday night.

First there was Miami's 21-0 run in the third quarter that sapped the energy from an otherwise raucous State Farm Arena and put the Heat in position to lead by double-digits in the fourth.

"We were down big," said Onyeka Okongwu, who has filled in for injured center Clint Capela (knee) this series. "I'm looking up at the scoreboard, I'm like, 'We got to get this thing going.'"

Atlanta stormed all the way back to take a three-point lead on an and-one putback layup and free throw by Okongwu with 1:41 remaining. However, the Heat went ahead again in the final minute, 110-109, on a P.J. Tucker corner 3-pointer with 54.2 seconds left and things looked precarious for the Hawks once again.

After the Tucker make, the Hawks and Heat traded a pair of missed jump shots, leading to Bogdan Bogdanovic's defensive rebound with 9.8 seconds remaining.

"When we got that miss, down one, basically it was like, 'Go,'" Hawks coach Nate McMillan said. "And [Young] knows what to do with that basketball and was able to make a play on that last possession."

While some could have questioned McMillan's reluctance to call a timeout in the third quarter when the Heat ran off the final 16 points of their 21-0 spurt without McMillan signaling for a break, that same instinct served the Hawks in creating Young's game winner.

"There was no way I was calling a timeout," McMillan said. "It was really a perfect situation for us, getting back in transition and he saw a lane and he attacked. He didn't settle. He attacked the basket and was able to finish that. So that play we're going to take, really, just about every time."

Heat coach Erik Spoelstra credited Young (24 points, seven rebounds, four assists) for coming through when the Hawks needed it.

"Young got a little bit of an angle and he was able to make a tough runner," Spoelstra said. "He makes that look easy, but that's not the easiest shot going full speed, and with the touch. It felt like it hit every part of the rim."

Miami called timeout to set up its final possession, and Butler missed a fadeaway 3 at the buzzer.

"Missed the shot, as simple as that," said Butler, who finished with 20 points after exploding for a career playoff best 45 in Game 2. "I'll make the next one."

Atlanta improved to 21-3 in its last 24 home games -- including Friday and its play-in win over the Charlotte Hornets -- and it was a bit of personal redemption for Young who shot 1-for-12 in Game 1 and then had 10 turnovers in Game 2.

"I just wanted to come here and take care of the ball and at least give our team a shot on the offensive end," Young said after finishing 6-for-14 from the field with three turnovers in Game 3. "I just got to keep doing it. It's only one game, so I got to do it again."

The Hawks received key contributions off the bench from Okongwu (nine points on 4-for-4 shooting, six rebounds), Bogdanovic (18 points, eight rebounds, six assists, two steals) and Delon Wright (13 points on 6-for-6 shooting, five rebounds).

"I'll say it was like a sigh of relief," Wright said of the mood in the locker room after the game. "Losing two games to them, the pressure was on us to get one of these wins."

It's a pressure that McMillan hopes his team can continue to find ways to manage as they look to extend their already improbable postseason run.

"Our backs were against the wall and the one thing that we talked about was being calm in that fourth quarter," he said. "I thought our guys got back to calm. Took a deep breath, got back to calm."