Kovalchuk took a cross-ice pass from Patrik Stefan and shot it past Senators goalie Jani Hurme. It was the Thrashers' first victory since Dec. 28 against Toronto.
Stefan had three assists, and Bob Corkum and Yannick Tremblay also scored for Atlanta.
Martin Havlat had a goal and an assist for the Senators, who twice lost leads in less than 90 seconds.
Hnat Domenichelli's wraparound goal with 14:27 left in regulation sent the game to overtime.
Marian Hossa had given Ottawa a 3-2 lead by converting a perfect 2-on-1 in the third period, taking a pass from Todd White and easily beating Rhodes. It was Hossa's 17th of the season.
But Domenichelli tied it 1:27 later, skating past Senators defenseman Karel Rachunek and around the net before stuffing the puck past Jani Hurme.
Thrashers goalie Damian Rhodes, who improved to 2-10-1, started in place of Milan Hnilicka and stopped 32 of 35 shots. Hnilicka was pulled early in Tuesday night's 7-4 loss to Philadelphia after giving up four goals in the first 12 shots.
Rhodes saved his best for last, gloving Daniel Alfredsson's point-blank shot early in overtime. Rhodes had fallen to make a save, and reached up from his stomach to make the save.
Atlanta's first lead came on Tremblay's slap shot from near the blue line. He fielded a pass from Stefan after a face-off and shot it through several bodies past Hurme.
Havlat's goal just 1:37 into the game gave the Senators a 1-0 lead, but Corkum answered 1:15 later to tie it at 1.
Notes: Atlanta captain Ray Ferraro played in his 1,220th game and took over 45th on the all-time list. He had been tied with Eric Nesterenko. ... Rhodes played 3.5 seasons for Ottawa before becoming the first player in Thrashers' history June 18, 1999. ... The Senators fell to 4-2-0 all-time at Atlanta. ... The attendance of 10,721 was the second-lowest of the season at Philips Arena. The low of 10,689 was set Dec. 12 against Montreal.
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