Jason Spezza also scored twice and Craig Anderson made 28 saves for Ottawa, which has won three straight and four of five. The 21-year-old Karlsson added two assists and leads NHL defensemen with 57 points.
"We all contributed today, especially Andy and the way he played in goal," said Karlsson, who has 12 goals and 45 assists in his third NHL season. "Everyone did their job."
The Senators (31-22-8) moved within two points of first-place Boston in the Northeast Division and one point behind Pittsburgh and Philadelphia for fifth place in the Eastern Conference playoff race.
Karlsson opened the scoring 1:08 into the first period with his 11th of the season on a sharp-angle shot from the corner to the left of Islanders goaltender Kevin Poulin. Spezza, who has seven goals and eight assists in his last six games, made it 2-0 just 27 seconds later.
Islanders coach Jack Capuano then yanked Poulin for Al Montoya but Ottawa kept pouring it on.
Karlsson made it 3-0 at 14:03 of the first with a low shot from the slot. It was the fourth career two-goal game for Karlsson, Ottawa's first-round draft pick in 2008.
"My goal is to keep working hard," he said. "A lot goes into being successful for 82 games plus the playoffs in this league. I'm still learning all the time."
Phillips added a power-play goal at 9:37 of the second and Lee posted his first of the season at 9:49 of the third.
"We are focusing well and playing playoff-type hockey," said Phillips, who played his 1,000th NHL game earlier this month. "It's what we need to do this time of year."
Spezza completed the scoring at 10:57 with a power-play score for his team-leading 27th goal of the season.
"We had the puck a lot today and we have confidence on offense," Spezza said. "When you are scoring goals, that only adds to that confidence."
It was the third shutout of the season for Anderson (28-19-6), who joined Ottawa in a trade with Colorado a year ago. The 30-year-old Anderson also moved into a tie with Henrik Lundqvist of the New York Rangers for fourth place in goalie wins this season.
"It was a total team effort today," Anderson said. "We have to keep this going."
Ottawa improved to 16-11-5 on the road. It was the Senators' most lopsided win of the season.
Ottawa also won 4-0 at Tampa Bay last Tuesday and 6-2 at Florida on Thursday.
The inconsistent Islanders (25-26-8) squandered their fourth chance in their last six games to move above the .500 mark for the first time since October.
"This was very disappointing. We came with no physicality and stood around and watched their best players," Capuano said. "You have to battle and compete and we didn't."
New York remained six points behind Toronto for the conference's eighth and final playoff spot.
The Islanders, who also lost 6-0 at home against Boston on Nov. 19, were seeking consecutive victories for the first time since winning 2-1 in overtime at Ottawa on Feb. 3.
"There wasn't any intensity in our game," Islanders defenseman and captain Mark Streit said. "We've had a couple of bad starts, but we topped it today. We keep doing this to ourselves."
The Islanders had defeated the Carolina Hurricanes 4-3 at home on Saturday. They are still 9-5-2 in their last 16 games and 14-9-2 in their last 25 contests, climbing back into the playoff race. They haven't reached the postseason since 2007.
New York is at Buffalo on Tuesday while the Senators return home to play Washington on Wednesday.
The Senators have won all nine games in 2012 in which Spezza has scored. ... The Islanders had won three straight over the Senators. ... The Islanders are 19-40-14 all-time against the Senators. ... Senators captain Daniel Alfredsson played his 1,111th NHL game and added his franchise-best 658th assist. ... Matt Moulson extended his games-played streak to 223. He hasn't missed a game in his Islanders career.
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