Earning consecutive wins for the first time since last Feb. 15 at New Jersey, Atlanta has just 24 points in its last 35 games.
The Thrashers, with only 10 points in 13 games this season, began the night as the worst team in the Eastern Conference, but the victory moved them a point ahead of the Islanders.
Anderson, a first-time NHL coach who won five titles in 13 years in the minors, allowed himself a few minutes to enjoy two straight wins.
"Yes, it feels good to come here and smile two times in a row," Anderson said before adding that his players had disappointed him in the second period. "That's not what I want this team to be about."
New York, coming off consecutive victories, failed to hold a 3-1 lead that Frans Nielsen and Trent Hunter provided with a pair of goals in a 17-second span midway through the second period.
"After our first two goals, we stopped skating," Islanders coach Scott Gordon said after watching his team fail to hold a third-period lead for the third time in four games. "All of a sudden, we were a different team. We were not aggressive. It was a little bit of old habits. We controlled play in the second period, probably the first period, too."
Kozlov and Jason Williams each scored in the first 1:17 of the third to force a 3-all tie before Enstrom made it 4-3 with a slap shot outside the left circle.
The goal was Enstrom's first since March 21 at Washington.
"Hopefully we can get back on track," Enstrom said. "It means a lot to the team to get two points in this game. It's always fun to get your first goal as long as we win."
Kozlov, with five goals and an assist over the last four games, forced a 1-all tie with 4:17 into the first.
Jon Sim, who tapped in his own rebound at the 2:23 mark of the first on a power play, opened the scoring with his third goal.
Ondrej Pavelec improved to 2-0-0 after stopping 24-of-27 shots to win his second straight start in replacing injured Atlanta goalie Kari Lehtonen. New York had a two-man advantage in the final 10 seconds after Chris Thorburn was penalized for hooking, but the Islanders managed just one shot, a 40-footer from Mark Streit that sailed over the net.
"We don't position ourselves properly in the offensive zone on the forecheck," Gordon said. "The fact that they play with caution when they get the green light is frustrating."
Joey MacDonald dropped to 4-5-1 for New York after stopping 23-of-27 shots. He confirmed that Enstrom's goal glanced off his skate and into the net.
"It's disappointing," MacDonald said. "It's disappointing. No lead is safe in the third period."
Nielsen's second goal this season, a tip-in from the right side of the crease, gave the Islanders a 2-1 lead 9:40 into the second. Hunter's fifth goal made it 3-1 after he beat Pavelec from the top of the slot.
Bryan Little passed from the rear boards early in the third to Kozlov, who scored on a backhander from the near the bottom of the left circle.
Williams tipped in his third goal on the power play after MacDonald stopped Enstrom's shot early in the third.
Atlanta defenseman Mathieu Schneider missed the game because of a hip flexor injury. General manager Don Waddell said Schneider isn't expected to join the team on a two-game road trip to Buffalo and Carolina. Atlanta called up defenseman Boris Valabik from AHL Chicago to add depth. ... Kozlov has 12 goals and 31 points in 36 career games against the Islanders. ... Hunter, in 17 career games against Atlanta, has eight goals and seven assists.