Anthony Beauvillier and Matt Martin also scored, and Ilya Sorokin stopped 26 shots to help the Islanders get their fifth win in seven games (5-1-1).
"The (penalty kill) showed up again," Cizikas said. "We did a good job of keeping them to the outside, not really giving them too much. ... We got a lot of really big saves when we needed them."
New York coach Barry Trotz was back on the bench after missing three games, one following the death of his mother and the last two in COVID-19 protocols.
Sorokin had eight saves in the first period, 12 in the second and six in the third to improve to 6-0-1 with a 1.50 goals-against average in seven career games against the Flyers. He is 10-7-5 on the season.
"Ilya, when we needed him, he stepped up and he was outstanding," Trotz said. "Good to see Beau and Nellie get on the scoreboard. We need them. ... And then Casey's line - the Identity Line has to have an identity, and today they did that."
Travis Konecny scored and Martin Jones finished with 27 saves for the Flyers, who are 0-6-2 during their losing streak.
"I thought we played pretty well," Jones said. "We were pretty good defensively. We were right there. ... We just have to find a way to win right now."
The Flyers went 0 for 4 on the power play, falling to 2 for 15 over the last six games.
"I think there was good and bad," Konecny said. "The first unit had a few looks and zone time. We made mistakes that gave them momentum. We have to clean that up."
The Islanders led 2-1 after two periods, and Beauvillier added to the advantage as he knocked in a loose puck in front at 4:44 of the third. Nelson had an assist on the play for his first multipoint game since he scored twice against Calgary on Nov. 20 in the first game at UBS Arena. He had one goal and one assist while playing in seven games since then.
The Flyers pulled Jones for an extra skater with about 2:40 left and James van Riemsdyk hit a goalpost a minute later.
Martin sealed it with an empty-netter with 1:23 left for his first goal of the season.
Sorokin made a stellar save, sprawling to his right to deny an attempt by Scott Laughton on a 2-on-0 break in the opening minute of the second period.
"I just work on the puck. I wait when the player shoots or gives a pass and he made a shot in the middle of the net and I was in good position," Sorokin said.
Jones saved a slap shot by Nelson at 5:15 and then stopped Kieffer Bellows' wraparound two minutes later and a backhand in close by Zach Parise less than 30 seconds after that to keep it scoreless.
Twenty seconds after the Islanders killed off their third penalty of the game, they got on the scoreboard when Nelson got a drop pass from Josh Bailey, paused and fired a shot past Jones into the top right corner for his 11th of the season with 8:16 remaining in the middle period.
The Islanders made it 2-0 less than three minutes later when Cizikas deflected Robin Salo's point shot with 5:32 left to end a 43-game scoring drought with his first goal since last April 1. Salo got his second NHL assist.
"It's been a long time coming," Cizikas said. "When I turned around and saw (Martin's) hands in the air I thought it hit him and went in."
Konecny got credit for his sixth of the season and 100th of his career when his dump-in toward the corner deflected off Islanders defenseman Noah Dobson's arm and into the net with 2:48 remaining to pull the Flyers within one.
Sorokin made a blocker save on a breakaway by Joel Farabee with a minute remaining to preserve the Islanders' lead.
The Islanders outshot the Flyers 13-8 in a fast-paced first period, including 6-1 over the last six minutes. Philadelphia's best scoring chance came when Laughton hit a goalpost 6 1/2 minutes into the game.
Islanders D Zdeno Chara played in his 1,636th career game, passing Scott Stevens for second-most by a defenseman in NHL history. Chris Chelios is tops on the list at 1,651 games. ... Flyers F Keith Yandle played in his 960th consecutive game, moving within four of tying Doug Jarvis for the longest streak in NHL history.
The Flyers dropped to 2-16-2 when giving up the first goal and 1-16-3 when trailing after two periods. ... The Islanders are 10-4-4 when scoring first and 11-1-1 when leading after 40 minutes. ... New York is 11-2-2 when scoring at least three goals. ... The Islanders are 6-1-1 in their last eight games at UBS Arena after starting 0-5-2 in their $1.1 billion new home.
Flyers: Host the Islanders on Tuesday night to open a two-game homestand.
Islanders: At Philadelphia on Tuesday night before a stretch of seven straight at home.
Matt Martin gets his first goal of the year with the empty-netter
Matt Martin gives the Islanders an insurance goal in their 4-1 win over the Flyers.