Spezza scored twice and Anderson made 45 saves as the Senators topped the Islanders 4-1 on Tuesday night.
The Senators (34-31-14) will still miss the postseason for the first time since 2011 and just the third time since 1996. They were making their first appearance this season at Nassau Coliseum, and the Islanders won the previous two meetings in Ottawa.
"It feels good to put together a team effort and come away with the win," Anderson said. "We played well."
Matt Kassian scored an empty-net goal at 19:48, his first this season. Spezza, the Senators' captain, had the chance for a hat trick, but instead passed to Kassian.
Frans Nielsen scored the lone goal for the Islanders with his career-best 25th in the team's home finale. Anders Nilsson made 28 saves for the Islanders (31-37-11), who won't reach the playoffs for the sixth time in seven years. They finished 13-19-9 at home.
"The effort has been there and will continue to be there," Islanders defenseman Travis Hamonic said. "We're going to keep pushing and playing hard until the end."
Spezza opened the scoring on a tip-in at 12:45 of the second.
Spezza's go-ahead goal at 12:45 of the third came when his shot edged under crossbar behind Nilsson. Ales Hemsky, who assisted on both of Spezza's goals, added his 13th at 14:26 of the third to make it 3-1.
The Islanders qualified for the postseason last year for the first time since 2007 and lost to the top-seeded Pittsburgh Penguins in a tough six-game opening-round series. But they never found firm footing this season after trading captain John Tavares' longtime linemate Matt Moulson to Buffalo for Thomas Vanek on Oct. 27. They subsequently traded Vanek to Montreal at the March 5 deadline.
The Islanders also had to contend with a litany of injuries to key players, particularly veteran defenseman Lubomir Visnovsky, who missed most of the season with a concussion and starting goaltender Evgeni Nabokov. Then, Tavares suffered a season-ending knee injury during the Winter Olympics in February.
The Islanders have dressed at least nine rookies in their last nine games in which they've gone 5-2-2. They are 7-4-2 in their last 13 contests. There are 11 rookies on the current roster: Defensemen Calvin de Haan, Scott Mayfield, Matt Donovan and Czuczman in addition to forwards Brock Nelson, Brett Gallant, Mike Halmo, Anders Lee, John Persson and Ryan Strome plus Nilsson. All but an injured de Haan played against the Senators.
"I'm proud of the young guys, they are playing hard," Islanders coach Jack Capuano said. "When we get our team back next year and play the way we can play, I think we can make a push and get back to the dance."
Gallant, 25, made his NHL debut. He had 245 penalty minutes in 56 games for Bridgeport of the AHL. The feisty forward leveled Kassian with a single punch midway through the first period.
"He caught me with one," said Kassian, sporting a postgame bruise under his right eye. "Sometimes that happens in a fight, you get punched right in the face."
The Islanders conclude their season with road games at Montreal, New Jersey and Buffalo. Ottawa hosts New Jersey and Toronto before finishing at Pittsburgh.
Islanders forward Cal Clutterbuck left during the first period with an upper-body injury and didn't return. ... Ottawa was without defensemen Colin Greening and Jared Cowen with lower-body injuries, plus forward Bobby Ryan (sports hernia). ... Defenseman Thomas Hickey is the only Islander to play all 79 games this season. ... The Islanders are 25-10-4 when they score at least three goals and 6-27-7 when they do not.