Andy Hilbert and Sean Bergenheim scored for the Islanders.
Weekes outdueled Joey MacDonald in a battle of backup goalies. Martin Brodeur, the Devils' four-time Vezina Trophy winner will be sidelined until March while the Islanders' Rick DiPietro will miss several more weeks with a knee injury.
Weekes got his first start since giving up four goals in two periods in a 5-2 loss to the New York Rangers on Nov. 12. Scott Clemmensen had started the last three for the Devils.
Clarkson got the Devils' first goal with 1:19 remaining, a power-play deflection of Johnny Oduya's hard shot from the right point.
Zajac struck moments later, deflecting Langenbrunner's shot. The changeup fooled MacDonald, and the Devils skated to the locker room with a 2-0 lead.
The Islanders got two outstanding saves earlier in the period, which prevented the game turning into a Devils' rout. MacDonald made the first stop, and defenseman Chris Campoli had the other.
MacDonald stopped Dainius Zubrus' drive from the slot with a glove save. On the second shot, MacDonald was out of position, having skated out of the crease to deny a breakaway bid Gionta. The puck wound up on Patrik Elias' stick with a wide open Islanders net. Campoli blocked the shot, maintaining the scoreless deadlock.
Both teams scored twice in the second period, giving the Devils a 4-2 advantage.
Hilbert scored for New York at 3:29, flipping a shot toward the net that appeared to hit a New Jersey defender. Salvador countered for New Jersey only 23 seconds later with his first regular season goal as a Devil. The defenseman, acquired late last season from St. Louis, scored once in the playoffs.
Bergenheim cut the Devils lead to 3-2 at 13:04, backhanding the rebound from Andy Sutton's shot that hit the post.
Gionta restored the two-goal margin for New Jersey with a short-handed goal with 15.4 seconds remaining.
In the third, Martin converted a 5-on-3 power-play advantage to put the Devils up 5-2 at 12:32.
Notes: Brendan Witt returned for the Islanders after missing four weeks because of a sprained knee. ... The Devils assigned Nicklas Bergfors to Lowell (AHL). He scored one goal in 8 games. ... Barry Tallackson made his season debut for the Devils following surgery to correct an irregular heartbeat, while John Madden sat out with a foot injury ... The Islanders started a three-games-in-four-nights trip that takes them to Buffalo Saturday and Montreal on Monday. ... Former Devil Bill Guerin played his 100th game as an Islander. ... Salvador and Jackman fought in the closing seconds of the second period. Jackman had a cut under his right eye. ... Frans Nielsen had to be helped from the ice early in the third after being laid out by an elbow from Devils defenseman Mike Mottau.