Mike Sillinger, Marc-Andre Bergeron and Chris Simon also scored for the Islanders, who have won five of six and recorded points in 17 of 19.
The victory moved the seventh-place Islanders within a point of Atlanta in the Eastern Conference playoff race, where four points separated fifth through eighth places entering Saturday. New York pulled two points ahead of eighth-place Carolina.
Smyth was acquired by the Islanders from Edmonton at the trading deadline earlier this week to bolster their post-season chances. New York already had the game in control when Smyth scored his 32nd goal with 2:54 remaining in the third period to make it 6-2.
Brian Pothier and Brian Sutherby tallied for Washington, which has dropped nine of 10 games and lost for the sixth straight game at home.
The Capitals struggled to create offence with a shuffled, injury-depleted lineup. After dealing away Richard Zednik, Dainius Zubrus, Jamie Heward and Lawrence Nycholat before the Feb. 27 deadline, the Capitals lost Boyd Gordon (sore knee) and Matt Pettinger (broken right thumb) in Thursday night's loss to Tampa Bay.
Power-play goals by Sillinger and Satan gave the Islanders a 2-0 first-period lead.
Sillinger swept in a shot from the left circle that caromed in off the right leg of goaltender Brent Johnson at 10:43. Satan's goal came with 1:32 left in the opening period, a right-post stuff after Sean Hill's slapper from centre-point had trickled through Johnson.
A double-minor against Donald Brashear erased a Capitals power play and set the stage for Bergeron's man-advantage goal 7:05 into the second. From the right point, Bergeron one-timed Tom Poti's cross-ice pass, clanging in his 10th goal off the left upright. The Islanders matched a season high with three power-play goals.
With 3:34 remaining in the second, Simon's goal put the Islanders up 4-0. From behind the goal line to the left of the net, Simon squeezed a shot between Johnson's leg and the post.
Satan scored his second goal - 23rd of the season - five minutes into the third period on a shot from the left circle. Pothier and Sutherby scored 24 seconds apart later in the period.
Notes: Islanders G Rick DiPietro made a career-high 14th consecutive start. ... New York played without D Bruno Gervais (ankle), C Shawn Bates (abdominal surgery) and C Alexei Yashin (knee). ... Zednik assisted on Satan's second goal, his first point in three games since the Feb. 26 trade. ... Washington's Alex Ovechkin had a three-game goal-scoring streak halted.