Sillinger scored from the left circle during New York's second power play in overtime.
"Anytime you get two power plays in overtime, you have to score," Sillinger said.
"We were desperate for a win," DiPietro said. "We haven't been playing our best hockey, so to come into Tampa, it's a good win for us."
"We battled," Richards said. "It was a great effort. We got one point. We would have liked to have two, but we couldn't get that done."
Hilbert gave New York a 2-1 lead at 10:45 of the third. Richards got the Lightning even with his second of the game during a 5-on-3 power play with 6:21 left in regulation.
Guerin put the Islanders up 1-0 at 12:09 of the second. He scored from the right circle after Mike Comrie won a faceoff in the Tampa Bay zone.
The Lightning tied it on Richards' power-play goal from just inside the blue line at 3:51 of the third.
"I think we played hard, and battled back a couple times," Tampa Bay coach John Tortorella said. "We did a lot of good things."
"We know the type of goaltender we have," Islanders coach Ted Nolan said. "He's going to make some big saves."
New York failed to convert on two short-handed breakaways during the first. Sillinger had the puck poked away from behind by Lecavalier, and Holmqvist made a glove save on Richard Park's shot.
The Islanders have scored three goals or fewer in 18 consecutive games.
DiPietro bailed out his teammates early in the second when he stopped Richards' breakaway shot during New York's 5-on-3 power play. He also made a save on Mathieu Darche, who skated alone down the slot with 13 minutes left in the period.
"He's the reason why we have 14 wins (this season)," Sillinger said. "He's the backbone of our team. The best penalty killer. He demands a lot from himself. He's an elite goaltender in the league."
Guerin's slump started after a goal against Tampa Bay on Nov. 1. He had just one assist during his scoring skid. ... Islanders LW Ruslan Fedotenko didn't play because of a shoulder injury. He is day to day. ... Lightning D Dan Boyle (wrist surgery) is making progress, but there is no timetable for his return. ... Lou Jankowski, who played for Detroit and Chicago from 1950-55, was at the game. His son, Ryan Jankowski, is the Islanders assistant GM and director of amateur scouting.