After winning Wednesday night in a shootout at Madison Square Garden, the Islanders completed a sweep of the home-and-home series.
"It was a tremendous lift for us," Parrish said. "If you can't get up for an Islanders-Rangers game, regardless of where you are in the standings and all those other things, you shouldn't be playing."
Miroslav Satan and Jason Blake also scored for the Islanders, who beat the Rangers 3-2 a night earlier. Rick DiPietro stopped 26 shots in the rematch to give the Islanders their first winning streak of the season.
"We came out these last two games with fire in our bellies," DiPietro said. "These last two games have been pretty intense, but at the end of the day, they don't ask you how you won, but if you won."
The Islanders were swept by the Rangers in six games in 2003-04, and before Wednesday were 0-8-2 in their previous 10 meetings since a 3-1 victory at Madison Square Garden on Nov. 23, 2002.
Jagr had his first three-goal game since Feb. 4, 2003, when he was with Washington. He leads the NHL with 10 goals and has a six-game point streak (8 goals, 2 assists).
Weekes, who missed the previous four games with a groin strain, was activated before Wednesday's game when he served as the backup to rookie Henrik Lundqvist.
The Islanders wasted little time as they put up a 3-0 lead in the opening 2:47 on their first three shots.
Satan scored 89 seconds with a wrist shot that beat Weekes high. Alexei Yashin assisted on the goal, extending his point streak to five games (3-3-6).
Just 26 seconds later, Blake made it 2-0 with another wrist shot. Parrish scored 52 seconds after that on the Islanders' first power play, tipping in a pass from Alexei Zhitnik.
"It wasn't a good start for us, for sure," Jagr said. "The whole game, we had to chase it. The only good thing being down 3-0 was that we had 55 minutes left to try to get at least a point."
Jagr scored twice in a 55-second span in the final minute of the period to bring the Rangers within 3-2.
He scored on a 5-on-3 advantage off a wrist shot with 57.7 seconds left, then notched his second goal with 2.1 seconds remaining by flipping a loose puck over DiPietro.
"I was still OK after 3-1, but that second goal was because we got lazy and didn't clear pucks the way they teach you how to do it in Pee-Wee," Islanders coach Steve Stirling said. "I liked our effort, but we have to do more work on execution."
Asham made it 4-2 when he broke in alone on Weekes, was pulled down, and still managed to get the puck in the net at 1:47.
"That was the turning point of the game for us," Stirling said. "If the Rangers score the next goal, it would probably have set us back on our heels a bit."
Rucinsky brought the Rangers back within a goal 44 seconds later on a backdoor play.
The Islanders went up 5-3 on a power-play goal by Parrish 51 seconds into the third period.
"The only difference between tonight's game and last night's game is that 40 guys were a little more tired tonight, and the building was different," Parrish said. "All I know is that if we play 82 games like this, I'm going to die."
Jagr made it 5-4 with a power-play goal with 5:42 remaining.
"It was a tough night to be a goaltender either way," Weekes said. "It's nights like this that make it frustrating to be a goalie."
Rangers coach Tom Renney wasn't pleased with his team's performance, even though this is only the second game this season in which the Rangers failed to earn at least one point.
"This is one of our less disciplined nights this season," Renney said. "We've been noted for gaining respect from our opponents, and we don't want that to slip away."
- The Islanders were 387 tickets short of a sellout, the first time a game between the local rivals wasn't filled to capacity at Nassau Coliseum since April 4, 1998.
- The Rangers had to evacuate the Garden City Hotel early Thursday afternoon after a chemical leak was discovered.
- The teams have played back-to-back games 19 times. The Islanders have swept twice, the other coming on Nov. 21-22, 1981. The Rangers swept seven times, and each team gained at least one point the other 10 times.
- The Islanders were winless in five home games with the Rangers (0-4-1). Their previous victory was March 25, 2002.