Panarin, with two goals and three assists, had three or more points for the fourth consecutive home game, tying Andy Bathgate (Nov. 8-19, 1961) for longest such streak by a Rangers player.
Panarin previously had five points in a game on Dec. 8, 2017, when he had five assists for the Columbus Blue Jackets in a 5-3 win against the New Jersey Devils.
[WATCH: All Islanders vs. Rangers highlights]
He is third in the League with 67 points (26 goals, 41 assists) and first among forwards with a plus-23 rating.
"Every now and then he one-ups himself, and he did that again tonight," Rangers coach David Quinn said. "We talked about it when we signed him over the summer, everybody stands a little bit taller, they put their chests out a little further and they have a little more swagger, but I think he's given us a lot more than that."
Video: NYI@NYR: Panarin nets backhander on breakaway
Jesper Fast had three points (one goal, two assists), Adam Fox had a goal and an assist and Alexandar Georgiev (32 saves) got his first win since Dec. 28 for the Rangers (22-19-4), who won their fifth straight home game.
The Rangers and Islanders will play twice more in the next eight days, including at Nassau Coliseum on Thursday.
"If we want to achieve our goal we can't let many nights slip through our fingers," Quinn said. "That doesn't mean you're going to win every night, but you want to give yourself a chance to win every night, and lately we've been doing a better job of that."
Video: NYI@NYR: Fast nets Panarin's rebound from in front
Jordan Eberle and Brock Nelson scored, and Semyon Varlamov made 29 saves before being replaced by Thomas Greiss in the third period after the Rangers' sixth goal. The Islanders (27-13-4) have five wins in 13 games since Dec. 17 (5-6-2).
"The puck management was terrible," Islanders coach Barry Trotz said. "Their top guys were better than some of our top guys. I thought our level of battle wasn't where I expected it to be tonight. We got what we deserved."
Eberle scored 18 seconds into the game to give the Islanders a 1-0 lead.
Video: NYI@NYR: Eberle nets patient goal just 18 seconds in
The Rangers felt they got some energy from back-to-back fights at 2:13. Micheal Haley and Ross Johnston engaged in the first fight before the face-off, and Brendan Smith and Matt Martin followed with another fight again before the puck dropped.
Smith and Martin were assessed game misconduct penalties for having the second fight on the same stoppage, forcing the Rangers and Islanders to play the final 57:47 with 11 forwards each.
"Those definitely gave us a little jolt," Rangers defenseman Jacob Trouba said.
Fast scored off a rebound of Panarin's shot at 8:14 to tie the game 1-1. The Rangers outshot the Islanders 17-7 in the first period, including 16-4 after the fights.
Chris Kreider scored on a deflection of Panarin's shot at 5:06 of the second period to give the Rangers a 2-1 lead.
"We turned some pucks over, they came at us hard, had some pressure, capitalized and we weren't able to turn it back in our favor even after getting that first one right away," Nelson said. "We couldn't stay on them."
Video: NYI@NYR: Kreider tips in Panarin's shot from in close
Fox made it 3-1 at 11:11 with a shot from the bottom of the right circle.
Panarin scored on a breakaway 49 seconds into the third period to make it 4-1, and again at 8:54 to make it 5-1. Trouba scored a power-play goal at 10:15 to give the Rangers a 6-1 lead. Nelson scored on the power play at 12:48 for the 6-2 final.
"He had five points and he could have probably had eight points in terms of the chances they were creating," Trotz said of Panarin. "Just didn't have an answer for it."
They said it
"I haven't seen that many odd-man rushes against us in a long, long time. That's on us. We have some guys who want to play differently than we want to play or we need to play against a team like this. We just didn't do it very well tonight." -- Islanders coach Barry Trotz
"I think what impresses me most is I've played with pretty high-skill guys and he doesn't take unnecessary risks and give up defensive responsibilities. He's a pretty good defensive forward as well, which makes it even more impressive that he's able to put up the points that he does." -- Rangers defenseman Jacob Trouba on Artemi Panarin
Need to know
The Rangers are 9-1-0 in their past 10 games against Metropolitan Division opponents. Their next three games are against Metropolitan Division opponents. … The last Rangers forward who had five points in a game was Derick Brassard, who is now with the Islanders, on Jan. 25, 2016. … Panarin has 22 points (six goals, 16 assists) in the past nine games. … Islanders center Casey Cizikas didn't play because of a lower-body injury.
Islanders: Host the Detroit Red Wings on Tuesday (Nassau Coliseum, 7 p.m. ET; MSG+2, FS-D, NHL.TV)
Rangers: At the Islanders on Thursday (Nassau Coliseum, 7 p.m. ET; MSG+, MSG2, NHL.TV)
Video: Panarin records five points, Rangers defeat Islanders