Couture opened the shootout by slipping a backhander between the pads of Dwayne Roloson. P.A. Parenteauthen shot high before Joe Pavelski was stopped.
Antero Niittymaki made a beautiful glove snare of a Frans Nielsen high backhander before Dan Boyle sealed the win as he lifted a backhander past Rolson.
Boyle scored in regulation while Niittymaki stopped 24 shots for the Sharks, who had dropped three of four coming in.
Parenteau netted a goal and Roloson stopped 37 shots in the loss for the Islanders, who have dropped nine straight for the first time since losing 10 in a row from December 4-23, 2008.
After a scoreless first period, Parenteau gave the Islanders the initial lead as he skated around from the left boards to the slot and zipped a wrister over the glove of Niittymaki at the 4:51 mark.
Skating with a 5-on-3 advantage, the Sharks tied the game when Pavelski fired a pass from the left circle to the top of the right circle where Boyle snapped a wrister through traffic with 5:34 left in the frame.
Niittymaki stopped 10 shots through the third period and overtime while Roloson made eight saves the rest of the way.
The Sharks continue their five-game homestand on Saturday against Calgary...The Islanders wrap up their three-game road trip in Los Angeles on Saturday...San Jose center Joe Thornton returned to play after serving his two-game suspension for an illegal hit versus the Blues on November 4...The Sharks have taken three straight and five of the last six against the Islanders.