"This is the first time I ever had four goals in any game, maybe except junior," Kozlov said. "It's not only me, but it's my linemates and my team."
Kozlov's wrist shot from the slot beat Kevin Weekes to the glove side with 6:08 left and he completed the scoring on another wrist shot with 1:36 left.
With their fifth victory in seven games, the Islanders pushed their points total to 31, one more than the Rangers and New Jersey Devils.
"It'll definitely make the bus ride back to Long Island a bit happier," DiPietro said, "but it's not like we've clinched the division. There's still a lot of hockey left."
Jaromir Jagr had the other goal for the Rangers in their first outright loss in seven games (3-1-3). Henrik Lundqvist started and stopped four of eight first-period shots. Weekes made 10 saves in the final two periods.
Just 12 seconds after killing off a penalty, the Islanders opened the scoring when Mike Sillinger fed Satan on the right wing. Satan moved to the lower right circle and wristed a shot high to Lundqvist's short side.
Satan, who led the Islanders with 35 goals last season, was off to a slow start, but has five of his seven goals in the last nine games.
The Islanders cashed in on their first power-play chance for a 2-0 lead. Point men Radek Martinek and Satan were playing catch when Martinek found Simon down low. His turnaround wrist shot banked in off the left post.
Kozlov put the Islanders up by three with 3:32 left on a backhander from the lower left circle. Asham stepped out of the penalty box and scored on a wrist shot from the right circle with 33.4 seconds left in the period.
"I don't think there were any soft goals," Lundqvist said. "At the same time, I am not satisfied with letting four goals in during the first period. I talked to Tom (Rangers coach Renney) and said a change is a good thing for the team."
Shanahan thought the Rangers weren't playing badly, though the scoreboard said otherwise.
"Believe it or not, we said after the first period that they were not playing as well as they thought they were and we were not playing as bad as we thought we were," Shanahan said. "We were still in the game, even though the score did not look like it. They have very few opportunities, and the opportunities they had, they scored on. We came out with some good energy, had some good shots, and before you know it, it was 4-0. We could have packed it in, but we didn't. We battled back."
Weekes replaced Lundqvist to start the second period, and passed his first test, stopping Shawn Bates with a pad save on a short-handed breakaway at 4:38.
Shanahan put the Rangers on the board with a short-handed goal at 6:45. On a 2-on-1 rush, Shanahan used teammate Matt Cullen as a decoy and fired a wrist shot between DiPietro's pads.
Jagr picked up his 13th goal at 16:22 on a wrist shot and, 66 seconds later, Shanahan knocked down a puck and quickly wristed a shot past DiPietro. Jagr has the most career goals against the Islanders in franchise history with 52.
Kozlov then scored his second goal of the night on a rebound at 18:41.
"We bent, but fortunately, we didn't break," DiPietro said. "They're a good club, and anytime you have a Shanahan and a Jagr and (Martin) Straka and (Michael) Nylander out there, they're dangerous."
Nylander had a goal disallowed with 64 seconds left in the period because the referee's whistle blew before he was able to jam a loose puck home.
Shanahan picked up the hat trick at 7:01 of the final period on a power play. He found a loose puck at the left crease and slammed it home.
"That was actually the turning point of the game," Islanders coach Ted Nolan said. "All of a sudden, a light went on in our brains, and we started getting our composure back."
Straka earned an assist on the goal, extending his points streak to seven games (6-7-13).
Kozlov said a 4-0 lead was tough to hold.
"It's not easy," Kozlov said. "In the back of our head, we started to relax a bit. Thankfully, we didn't relax so much."
It was the first hat trick of the season for both Shanahan and Kozlov, and Kozlov's second career hat trick. ... Simon was clipped with an accidental high stick with 8:58 to go. He was helped off the ice with a towel covering his face. Ranger fans found a new target to direct their booing toward: former Blueshirt defenseman Tom Poti, who has played well for the Islanders. ... The scheduling for the eight games between these two rivals features the oddity of the Rangers hosting four of the first five meetings. If last season was any indication, it wouldn't matter to the Rangers, who won three of four games at Nassau Coliseum.