In eight shootouts, Satan has scored six times -- including three against Brodeur.
"He looks like he's not even trying out there and he just picks me apart every time, so it's definitely not a fun situation," Brodeur said.
Satan matched Viktor Kozlov's goal in the first round, and Hunter then gave the Islanders the lead. Rick DiPietro stopped U.S. Olympic teammate Brian Gionta and Jamie Langenbrunner -- New Jersey's lone scorer in regulation -- to give New York its fifth win against the Devils in seven games (5-1-1).
Mark Parrish, the subject of trade rumors, got the Islanders even at 1 early in the third period. The Devils fell to 16-4-3 since Dec. 31 and have helped New York maintain its playoff hopes.
"I didn't think they played that great," said Brodeur, 6-3 in shootouts. "They didn't show me that they are a desperate team. Their intensity was not as much as what we have to go through against the Rangers or the Flyers."
DiPietro finished with 30 saves and improved to 6-3 in shootouts. Brodeur stopped 29 shots.
New Jersey carried a 29-27 shots advantage into the extra session and then received its ninth power play -- compared to two for New York -- at 1:35 when the Islanders had too many men on the ice.
Langenbrunner nearly won it, but his drive rang off the right post on New Jersey's best scoring chance with 2:22 left.
Brodeur then rescued the Devils without a stick on New York's next trip up ice. Satan, who scored the decisive goal against Brodeur on Thursday night in a shootout, hit the goalie in the pads as time ran out.
"I just have a hard time stopping him in breakaways," Brodeur said. "I stop it all night long during the game."
New Jersey took the play to the Islanders in the second period after being outshot 10-5 in the first. The Devils benefited from having the first six power plays of the game but only took advantage once. They even had a five-on-three power play after Langenbrunner's goal.
New York didn't record its first shot of the period until Arron Asham was alone in front at 10:05. Brodeur knocked the shot away with his glove. By then New Jersey held an 11-1 shots advantage.
But the Islanders bounced back again early in the third and put the pressure on before Parrish finally dented Brodeur to complete an end-to-end passing rush.
With New Jersey making a line change, Alexei Zhitnik cleared the puck from his zone with a long feed to Shawn Bates at the Devils' blue line. Bates sent a backhand pass to Parrish, who while in stride chipped it in from the bottom of the left circle to tie it at 4:44 on New York's fourth shot of the period.
Parrish's 24th goal of the season might be his last for the Islanders because he is a prime candidate to be dealt before Thursday's trade deadline.
"I didn't think of that at all during the game, but once we got back in here, the guys started giving me grief," said Parrish, who can be a free agent at season's end. "My name's been in rumors for a long time, so I'm kind of used to it, but it is in the back of my mind.
"If it was my last game here, I'm glad we wound up putting on a good show," he said.
The Islanders are 12th in the Eastern Conference, seven points out of a playoff spot, and could look to be sellers. They viewed the four games before the deadline as a chance to prove they are still contender.
New York beat New Jersey twice and Philadelphia once but lost to lowly Washington on Monday.
"I'd like to maintain the team we have and see what we can do with it," coach Brad Shaw said. "I think we needed to be seven points away to realistically keep this team together.
"There were two or three games recently where we didn't show up, and those points would have helped us out a lot," he said.
Devils center Patrik Elias skated Tuesday morning but missed his fourth consecutive game because of bruised ribs sustained in the Olympics. New Jersey defenseman Richard Matvichuk didn't practice and sat out his third straight with a back injury. ... New York defenseman Brent Sopel didn't play because of a bruised knee.