The Hawks outscored the Celtics 16-4 in the second overtime.
Jeff Green had 17 points for Boston. Courtney Lee had 16 points and Avery Bradley had 14.
Boston appeared headed for a rout when it led 48-21 in the second quarter.
"The most important part is how we didn't lay down in the second half," Smith said.
"We didn't get discouraged. We came out in the second half and played outstanding. ... We applied a little pressure and took the fight to them in the second half."
Atlanta trailed 57-38 at halftime following a modest recovery.
The Hawks' comeback gained strength as they outscored the Celtics 21-2 to open the second half, including a 19-0 run for a 59-all tie. Korver had three 3-pointers in the run.
The comeback was Atlanta's biggest since at least the 2000-01 season, according to STATS LLC.
"This is probably one of the most amazing comebacks I've been associated with," said Hawks coach Larry Drew, who said he was including his years as a player and a coach.
It was Atlanta's third straight win following four losses in five games.
"This was a very gutsy win for us," Drew said. "It shows what we're made of. It shows we have the ability to persevere."
The Hawks led 80-77 after Korver's fifth 3-pointer, but the Celtics regained the lead on back-to-back baskets by Pierce and Courtney Lee.
Horford started after missing one game with a sore left calf, and Devin Harris had 14 points in a backup role after missing two games with a sprained left ankle.
NOTES: Milwaukee also rallied from 27 points this season to beat Chicago 93-92 on Nov. 23, 2012. ... The Celtics' last six-game losing streak was April 4-15, 2007 ... The 10 points were a season low for first-quarter points for the Hawks. ... Korver extended his streak of games with at least one 3-pointer to 37, the longest of career and longest active streak in the NBA. ... Hawks C Zaza Pachulia returned after missing one game with a sore right Achilles.