Larkin scored for the first time in 10 games to pull Detroit within 2-1 at 10:36 of the second period. The goal came 10 seconds into the Red Wings' fifth power play of the game.
Detroit finished 1-for-6 with the man-advantage.
New York responded with a power-play goal after Barzal drew two interference penalties in 32 seconds. The first penalty had just expired when Eberle scored his first of the season on a rebound at 16:34 to make it 3-1.
"If you stay 5-on-5 and play the way we're capable of, I felt very comfortable," Islanders coach Barry Trotz said. "But we took too many penalties and the dam finally broke. Our penalty killers did a really good job, but they make it 2-1 and finally we were able to get something at the end of the first 5-on-3."
Eberle had three points, all assists, in the first five games of the season, then missed 10 with a lower-body injury. He had two assists in nine games entering Monday.
"I felt like I've had a ton of chances since I came back from the injury, but nothing has gone in," he said. "You get frustrated and you start gripping the stick too tight, but you just have to keep working. It's a funny game; you go that long without scoring and then you get two."