Brock Nelson, Devon Toews, Anders Lee and Ross Johnston scored for New York (38-21-7), which moved into a tie with the Washington Capitals for first place in the Metropolitan Division and avoided a three-game losing streak.
Robin Lehner made 21 saves but left the game with 4:40 remaining in the third period after a collision with Brady Tkachuk. Thomas Greiss made three saves in relief.
WATCH: [All Senators vs. Islanders highlights]
Trotz joins Scotty Bowman (1,244), Joel Quenneville (890) and Ken Hitchcock (843) as the only coaches with 800 NHL wins.
"Least on my mind today, trust me," said Trotz, who didn't have an update on Lehner's condition and was scrambling for a lineup with a flu bug going through the Islanders dressing room.
Brian Gibbons, Oscar Lindberg, Zack Smith and Jean-Gabriel Pageau scored, and Craig Anderson made 34 saves for Ottawa (23-38-6), which has lost eight of its past nine games.
"I thought it was a really good game against a really good team," Senators coach Marc Crawford said. "They're not at the top of the standings by accident; they play the game the right way. The way that they support the puck really taught us some lessons."