Madison Square Garden has become a safe haven for the Islanders, who hadn't won anywhere since beating the Rangers there on Dec. 26. Despite not scoring on the power play again, the Islanders shook off the tying goal at the start of the third period and knocked off the Rangers for the fourth time this season and fifth in a row overall.
Sillinger gave the Islanders a 3-2 lead in the second period after the teams split four goals in the first.
Miroslav Satan and defenceman Brendan Witt scored for the Islanders, and Martin Straka and Jaromir Jagr had goals for the Rangers, who rallied from 1-0 and 3-2 deficits to tie.
Brendan Shanahan, playing with a bruised right foot, woke up the home crowd with a power-play goal 42 seconds into the final period, but the Rangers couldn't sustain the momentum and had their four-game winning streak snapped.
Jagr and Shanahan took the ice despite injuries and each broke out of their longest goal droughts of the season.
Jagr is battling a hip injury that forced him to miss the latter stages of the Rangers' win Saturday at Montreal, but he chipped in with his first goal in nine games and helped set up Shanahan and Straka.
Shanahan took a great pass from Jagr, who was guarded by Richard Park and Bruno Gervais along the right wall, and quickly fired a shot past Rick DiPietro for his team-leading 24th as Andy Hilbert served a holding penalty.
But the Islanders came right back, using the Rangers' sloppy play and quick shots to score on Henrik Lundqvist, who had allowed only eight goals in the winning streak.
Randy Robitaille slid a pass back to Kozlov, who ripped a shot from the high slot inside the left post to give the Islanders the lead for good at 5:17 of the third.
Before that goal could be announced to the crowd, which featured large pockets of Islanders rooters, Sillinger intercepted a pass from Rangers forward Petr Prucha. Sillinger let a shot go that Lundqvist blocked with his pad, but the rebound came right back out and Sillinger quickly pushed it in for his 14th goal this season at 5:44.
The closest the Islanders came to a power-play goal was when Witt scored his first goal in 78 games during a delayed-penalty call against the Rangers with 1:23 left in the second period. The Islanders went 0-for-3 on the power play and are 1-for-24 in their past seven games.
Notes: The Rangers held a 31-26 shots advantage. ... Islanders forward Shawn Bates missed his second straight game (hand). ... Rangers D Karel Rachunek practised Monday but sat out his second straight (bruised foot). ... Islanders captain Alexei Yashin had 16:02 of ice time in 20 shifts after being benched in the second period of a loss at Carolina on Saturday. ... Straka scored his 23rd goal, one more than he had all last season.