Jason Blake, Alexei Zhitnik and Viktor Kozlov also scored, Alexei Yashin added three assists and Rick DiPietro had 14 saves for the Islanders, who won their third straight game and ended the Devils' two-game winning streak.
Rookie Travis Zajac scored for the Devils and set up another by Zach Parise. Martin Brodeur, who had posted shutouts in his last two starts, had 22 saves.
The Islanders went 6-2 against New Jersey last season and dominated the first of eight meetings this year, limiting the Devils to 10 shots in the first two periods.
Hunter, York and Satan set the tone after struggling recently for New York.
Hunter had a goal and an assist in the first 11 games. While York had two goals and three assists, the centre had not had an assist in five games. After scoring in the opening minutes of the season opener, Satan had not put the puck in the net again.
Hunter ended a seven-game goal drought by beating Brodeur with a snap shot from between from circles 8:00 into the first. York made the pass after picking up a loose puck in the Devils' zone.
After Zajac tied it with about five minutes left in the period, Satan needed only 32 seconds to put New York ahead for good. York took the puck from Scott Gomez in centre ice and set up Hunter for a shot in the left circle. Satan lifted the rebound over a prone Brodeur.
Zhitnik stretched the lead to 3-1 midway through the second with a shot from the point. Kozlov scored from the left circle about five minutes later off a great cross-ice pass by Yashin.
Parise narrowed the deficit to 4-2 by redirecting a Jamie Langenbrunner pass into the net late in the second.
Notes: Blake originally got credit for the Zhitnik goal until a scoring change was made in the third period. ... Jim Dowd, a member of the Devils' 1995 Stanley Cup championship team, re-signed with the Devils on Thursday. He last played for New Jersey in 1995-96. To make room for him, RW Dan LaCouture assigned to Lowell of the AHL after clearing waivers. ... Until Hunter's goal, Brodeur had gone 136 minutes, 28 seconds without giving up a goal.