"We didn't have a good first period at all," White said. "It's tough to get in the flow when you're killing penalties, but it was a great individual effort by Kovy to get us on the board."
Atlanta improved to 6-3 under interim coach Don Waddell, the team's general manager. The Capitals lost their 10th in 12 games after starting the season with three straight victories.
Kovalchuk scored his NHL-leading 13th goal, and seventh in three games, to make it 1-0 at the 13:40 mark of the second.
Three minutes later, Kovalchuk doubled over in pain against Atlanta's goal-line boards. He briefly lay face-down before skating off the ice without assistance but did not return.
Waddell said Kovalchuk will have an MRI on Wednesday.
"I can't really move or get up right now," Kovalchuk said. "But I hope it loosens up, and I hope to be ready for the next game."
Michael Nylander's fifth goal, on the power play with 15:40 remaining, made it 1-1.
White was at the left side of the crease when he beat goaltender Brent Johnson, who lost his third straight start. The goal was White's third of the season and gave him eight points in his first season since leaving Minnesota to sign a free-agent contract with the Thrashers.
Atlanta, which has won four of five, won consecutive games for the first time since the final two regular-season games last year. The Thrashers' next game is Friday at New Jersey.
Washington has scored just three goals in its last four games.
Thrashers goalie Ondrej Pavelec stopped Washington's first 25 shots before Nylander's wrister from the left side of the slot beat his stick side. Nylander had an uncontested shot after taking Viktor Kozlov's wraparound pass.
Pavelec, a rookie, has won his first two career starts. He made 31 saves.
"We had some great chances in the first period and in the third period," Capitals forward Alexander Ovechkin said. "We had more chances than they did."
Washington coach Glen Hanlon gave No. 1 goalie Olaf Kolzig the night off. Johnson dropped to 7-3-1 in his career against Atlanta, 3-2-1 at Philips Arena.
Ovechkin, the Capitals' leading scorer over the last two years and one of the NHL's top young players, went consecutive games without a point for the first time this season.
A former Calder Trophy winner as the league's best rookie in 2006, Ovechkin still believes that Washington will soon end its slump.
"We have experienced guys this year," Ovechkin said. "I have more confidence this year, but we have to figure out why we're not scoring more."
Thrashers forward Slava Kozlov, who assisted on White's goal, has seven points, including three goals, over his last five games. ... Atlanta improved to 3-0 in overtime after winning its first two in a shootout. ... If the game had gone to a shootout, Kovalchuk would have stayed off the ice. "I tried my best to come back," he said, "but the doctor said no." ... The Capitals are 0-1 in overtime this season. ... The Thrashers went 0-for-5 on the power play, dropping them to a combined 3-for-23 in their last five victories. ... Washington did not take a shot on goal in overtime.