New York has lost six straight, a stretch that has seen the Islanders outscored 29-11.
Spezza's second goal of the season came 6:17 into the second period and gave Ottawa a 3-0 lead. He assisted on the other three goals by Sergei Gonchar in the first period, David Hale in the second and Chris Kelly in the game's final 13 seconds.
Ottawa has won four of its last five after capturing just two of the season's first eight contests.
Brian Elliott, who has been filling in for the injured Pascal Leclaire, made 28 saves for the Senators.
Dwayne Roloson gave up three goals on 26 shots.