With rookie Al Horford guarding him at the top of the key, Durant pumped once to get the defender in the air before shooting over his outstretched arm.
"Al Horford was right on it, but I faded away, and fortunately it went in," Durant said. "I'm starting to get used to playing every other night or so, and I'm gaining confidence with my game."
Durant, the No. 2 overall draft pick, finished with 21 points to help Seattle improve to 2-8 following a victory on Wednesday night at Miami.
Joe Johnson had a season-high 39 points for the Hawks, who have lost three of four.
Atlanta's Josh Childress missed the first of two free throws at the end of regulation, but made the second to force overtime. The third-year guard had made his first 24 free throws this season before missing the potential winner.
"We had our chances in regulation and overtime," Hawks coach Mike Woodson said. "We missed free throws, we had a botched play on a defensive assignment where we gave up a 3 in the corner. It's doing the little things when you're talking about winning a close game like this."
Seattle's biggest lead was 14 early in the third quarter, but the Hawks came back several times in a game that featured 15 lead changes and was tied 20 times.
Atlanta's last lead came on Marvin Williams' basket on the first possession of the second overtime, making it 116-114. Wilkins hit a 16-footer on the right baseline with father Gerald Wilkins and uncle Dominique watching a few feet away.
Johnson was defending Wilkins tightly when his basket gave the Sonics a 120-118 lead.
The basket allowed Wilkins to atone for a mistake near the end of regulation, when he let an inbounds pass bounce away with the game tied at 107.
"Playing in front of family certainly made it special for me," Wilkins said. "The most important thing is that we got the win."
Childress, who scored 22 points before fouling out, wasn't the only Hawk to miss an important free throw. Horford, the third overall pick, watched the first of two foul shots bounce off the rim with 29 seconds remaining in the first overtime.
He made the second attempt to give the Hawks a 114-112 lead, but Wilkins' 5-footer tied it on the next possession, driving the left baseline and getting Horford to commit too early on defense.
After Atlanta called timeout with 23 seconds remaining, Salim Stoudamire missed a potential game-winning 3 as buzzer sounded to end the first overtime.
Durant, though, was the talk of both locker rooms. Sonics coach P.J. Carlesimo praised guard Wally Szczerbiak for making a nearly perfect pass to the rookie star.
"Wally made a good decision, and Kevin was able to get off the shot and hit it," Carlesimo said. "He'll probably do this many, many times before his career is over."
The Hawks were without F Josh Smith (quadriceps) and three guards -- rookie Acie Law (ankle), Speedy Claxton (knee) and Anthony Johnson, who left Philips Arena before the game for personal reasons. ... Atlanta started Tyronn Lue in Law's place. ... Seattle has eight consecutive wins against the Hawks, including five straight in Atlanta.