Teague saw his floater bounce off the rim with 0.9 seconds left in regulation, and he missed a baseline jumper at the first-overtime buzzer.
At the end of the second OT, Teague let the clock wind down, then fired a shot from the left of the key. The ball hit the rim, then the backboard, then the rim three more times before falling through the net.
He fell back onto the court in celebration and was swarmed by teammates.
Teague said the last shot was supposed to be a pick-and-roll.
The 6-foot-2 guard converted 14 of 24 shots from the field.
The Cavaliers (10-18) couldn't stop turning the ball over, as they matched their season high with 24 turnovers.
"Of course (we let this slip away), especially when we laid it on the line like that," Cavaliers guard Dion Waiters said. "We have to close it out, especially at home."
CLEVELAND suffered its third consecutive loss and fell for the fifth time in six games.
Irving committed six turnovers, including a costly one with 2.4 seconds left in the first overtime. Instead of going for a possible game-winner, CLEVELAND had to hope the Hawks' last shot didn't go in. Teague missed a 17-foot baseline attempt, sending the game into the second OT tied at 108.
Then, the Hawks converted a rare four-point play by Korver, when Anderson Varejao fouled the guard beyond the 3-point arc.
The Cavaliers just couldn't get the needed stops.
"It's tough, man," Waiters said. "We could have put them away a few times. We gave them a lot of second-chance points."
Cavaliers coach Mike Brown added, "I just felt we made too many mistakes as a group down the stretch of regulation and overtime and the second overtime -- not just offensively -- but defensively as well.
"We still don't understand that we can win games by getting stops. That's a big downfall of ours. Somehow we have to figure out we can't outscore everybody."
"It was sore," he said. "I didn't want to miss any more games."
However, he subsequently sank four consecutive shots and was off and running. In all, he made 17 of 33 attempts Thursday.
NOTES: Hawks F DeMarre Carroll (wrist/thumb) and C Al Horford (right shoulder) were knocked out of the game with injuries. Hawks coach Mike Budenholzer said they would be evaluated Friday. ... F Earl Clark replaced F Alonzo Gee in CLEVELAND's starting lineup, and he scored three points. Gee never got off the bench. ... Cavaliers G Kyrie Irving remains second in the fan voting for the NBA All-Star game among backcourt players in the Eastern Conference. Heat G Dwyane Wade is first with 565,455 votes, followed by Irving with 524,000. Irving has a substantial lead over injured Bulls G Derrick Rose, who has 299,950 votes. ... Cavaliers G C.J. Miles knows shooters, and he holds Hawks G/F Kyle Korver in high regard. "He might go down as the best shooter ever," Miles said. ... Hawks G John Jenkins is sidelined due to a back injury, while Atlanta F Cartier Martin was out because of a sprained ankle.