Poulin was yanked after allowing scores 27 seconds apart to Karlsson and Jason Spezza in the first 1:35 of action.
"There was zero intensity in our game," Islanders defenseman Mark Streit said. "We can't start a game like that, down two goals right away. We didn't find a way to respond and get back in the game."
Craig Anderson stopped 28 shots to notch his third shutout of the season for the Senators, winners in three straight and four of five.
Al Montoya finished the contest by giving up four goals on 27 shots for the Islanders, who have alternated losses and wins in their last nine games.
Things didn't look any brighter for the hosts after Montoya spelled Poulin, as Karlsson made it 3-0 just after a power play expired by wristing a shot from the slot at 14:03.
Phillips ripped a shot home from the right point on another Ottawa advantage at the 9:37 mark of the second period.
Lee struck for his first goal of the year just prior to the midway point of the third on a long slap shot from the left point, and Spezza made it 6-0 on a power-play rocket from the left circle just over a minute later.
"We got two days rest. Not many teams get that this time of year," said Spezza. "And there's a benefit to playing games early on, because we can get some energy to move forward."
Anderson posted his second whitewash in three starts, after turning in a 4-0 decision at Tampa on February 14...The Islanders were blanked for the ninth time this season...New York forward P.A. Parenteau, who was given a minor for unsportsmanlike conduct in the first period after protesting a slashing call, was given a game misconduct following Ottawa's fifth goal for arguing a non- call...Lee's last NHL goal came against the Rangers on November 14, 2009...Karlsson's four points gave him 57 on the season, six shy of the Senators' record for defensemen set by Norm Maciver in 1992-93.