Even with Al Horford's season-ending injury in December and a 1-14 stretch in February and March, the Hawks made the Eastern Conference field with two games remaining.
"I'm happy for all the guys," Teague said. "We worked so hard all year. Injuries and everything that can happen to a team happened. We were able to still make it to the playoffs."
Atlanta's third straight victory eliminated the New York Knicks from playoff contention. Williams scored the Hawks' last 12 points.
The two-time defending champion Heat, who have lost three of four, gave up the ground they gained in the standings Friday by beating Indiana at home and taking a one-half game lead over the Pacers.
Miami and Indiana are now tied atop the Eastern Conference with 54-26 records. The Heat's 85 points were their third-fewest this season.
"It's not finished," James said. "I see the standings. There's a couple games left and the seeding is changing every day."
James and Chris Bosh returned to the floor with 6:42 remaining and Miami trailing 81-72, but DeMarre Carroll stole the ball from Bosh and Mike Scott's fastbreak dunk put Atlanta ahead by 11 with 6:09 remaining.
Heat coach Erik Spoelstra called timeout to settle down his team, but James missed a jumper and Teague hit a free throw to make it 84-72.
James hit a 3 two possessions later, but Teague drove to the rim again for an 11-point lead, and Miami never got closer than eight the rest of the way.
Wade finished the first half with 16 points, going 7 for 7 from the field in just 11 minutes. He began the third quarter with an offensive foul, a missed jumper and hit one of two free throws, but he finished strong before sitting out the final 5:46 to rest.
"It was good, better than I can envision," Wade said. "When you're out so long, you're just thinking of getting your wind back. I think the best thing that happened was the first play of the game, getting that block and then LeBron kind of holding up and letting me get the first dunk. From there, I kind of was into the game."
The Hawks never trailed after Teague's 12-footer made it 53-51 early in the third.
Carroll finished with 17 points, and Paul Millsap added 11. Scott scored 12 points in a reserve role, joining Williams to help Atlanta's bench outscore Miami's 35-8.
Bosh finished with 11 points.
"Needless to say that was a tough second half," Spoelstra said. "We couldn't get it going on either side of the court. It's disappointing to have this type of performance on the road."
Miami visits Washington on Monday and end the regular season at home Wednesday against Philadelphia. The Pacers host Oklahoma City on Sunday and visit Orlando on Wednesday to close the regular season.
Atlanta hosts Charlotte on Monday and visits Milwaukee on Wednesday. As the No. 8 seed, the Hawks will play either Miami or Indiana in the first round of the playoffs.
"I'm happy that we get to play more games and I get to talk more about improving and getting better each practice," Atlanta coach Mike Budenholzer said. "We want to build something here. Miami has been in the Finals for three years in a row. There are a lot of teams that have had a lot of success. It takes time to build your habits."
A moment of silence was held before the game following the death Friday of Lou Hudson, the six-time All-Star with the Hawks who retired in 1979 and still ranks as the franchise's third-leading scorer. ... The Hawks are wearing a black stripe on their uniform to acknowledge Hudson's passing. ... The Heat dropped to 22-18 on the road. ... The Hawks improved to 24-16 at home. ... Atlanta is in the playoffs for the seventh straight year, longest active streak in the East.