The Avalanche knew they were ready to play. The Thrashers, meanwhile, knew they were flat.
Anderson stopped 24 shots and Fleischmann had a goal and an assist to help Colorado end a five-game road losing streak.
Fleischmann said the Avalanche "were a desperate team" after four straight losses, including three in overtime.
"We didn't have a three-period game lately and we need it," Fleischmann said. "We came and did what we needed to do, and I think we played good."
Niclas Bergfors scored to give Atlanta a 1-0 lead 5:37 into the opening period, but the Avalanche were doing most of the shooting. Bergfors found the net on Atlanta's first shot at a time Colorado already had eight shots on goal.
The Thrashers picked up the pace in the third period, when they took a 12-5 advantage in shots on goal, but they still were outshot 29-26 for the game.
"We weren't ready for the game," said Thrashers goalie Ondrej Pavelec. "... We played a great third period, but it wasn't enough. We have to be ready to play next time."
With Colorado leading 3-2, Anderson deflected Ben Eager's shot in front of the net with about 2 minutes remaining, then denied the Thrashers on a power play in the final 1:24.
The Thrashers pulled Pavelec for the power play and gave up Paul Stastny's empty-net goal.
Greg Mauldin also scored for Colorado, which ended the Thrashers' home winning streak at six. The Thrashers were denied matching the longest home streak in franchise history.
Bryan Little also scored for Atlanta.
"It was a game we were ready to start," Colorado coach Joe Sacco said. "They scored early, but I still liked the way we started."
Fleischmann assisted Hejduk's go-ahead goal with a pass from behind the net.
"I saw the play behind the net and I thought [Fleischmann] was going to go around," Pavelec said. "He made a smart pass behind the net."
Pavelec stopped 25 shots. He took only his third loss in his streak of 11 straight starts.
The Thrashers, who face a league-high 34.9 shots per game, were outshot 16-5 in the first period.
"We didn't play great," Eager said. "We came out flat. We didn't have the best week of practice, and it showed tonight."
Atlanta coach Craig Ramsay said his team "wasn't as sharp" after a three-day break.
"It was the first time in a long time we had a few days off between games, and we didn't react well," Ramsay said. "As the coach, I have to take a lot of the blame. Whether we practiced too hard or whatever, we were just not ready."
The Avalanche scored two goals in the second period. Fleischmann scored a power-play goal early in the period to tie the game. On Atlanta's first power-play opportunity of the game, Mauldin scored a short-handed goal for a 2-1 lead.
The Thrashers pulled even on Little's goal late in the second period.
Anderson improved to 4-1-1 against Atlanta.
The Avalanche avoided matching the franchise record of six straight road losses from Jan. 2-29, 1992, when the team was based in Quebec City. Colorado's last road win came on Nov. 20 at Dallas.
Pavelec was 8-2 with a 1.29 goals-against average in the first 10 games of his stretch of 11 straight starts.
Ramsay plans to rest Pavelec and start Chris Mason on Saturday at the New York Islanders. ... The Thrashers won seven straight home games from March 2-18, 2007, on their way to the Southeast Division championship and the only playoff appearance in franchise history. ... F Brandon Yip returned to Colorado's lineup. He was a healthy scratch against Florida on Tuesday.