Anton Forsberg made 24 saves for the Senators (22-36-5), who have lost five of six.
"Both teams had chances," Ottawa center Chris Tierney said. "We had some good looks in the first, second, even the third. They found a way to get one. In a game like that, whoever scores first usually wins."
Mayfield gave New York a 1-0 lead at 9:49 of the third period, shooting short side after Nelson's centering pass was blocked by Dylan Gambrell. It was Mayfield's third goal of the season and first since Jan. 21 (28 games).
Lee pounced on a loose puck in front at 17:25 to make it 2-0, and Zach Parise scored an empty-net goal with 47 seconds left for the 3-0 final.
"Definitely a cloggy game for the first couple (of periods), and I think we played a better third period, a little more straightforward," Lee said. "We made an emphasis on really working in the [offensive] zone as much as possible and eliminated our turnovers."
Varlamov made 13 saves in the first period, including against Tierney on a breakaway at 14:03.
"To get a save like that after a little scrum, it's definitely big, it keeps us in the game," Islanders center Casey Cizikas said. "He played well all night."