Craig Anderson made 45 saves for the Senators, who will still miss the post-season 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.
Matt Kassian scored an empty-net goal in the closing seconds, his first of the season.
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.
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. Hemsky, who assisted on both of Spezza's goals, added his 13th of the season at 14:26 of the third.
The Islanders qualified for the post-season 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, and traded 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 defenceman 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: Kevin Czuczman, Calvin de Haan, Matt Donovan, Brett Gallant, Mike Halmo, Anders Lee, Scott Mayfield, Brock Nelson, Anders Nilsson, John Persson, Ryan Strome. All but de Haan played against the Senators.
Gallant, 25, made his NHL debut. He had 245 penalty minutes in 56 games for AHL Bridgeport. The feisty forward levelled Kassian with a single punch midway through the first period.
The Islanders conclude their season with road games at Montreal, New Jersey and Buffalo. Ottawa hosts New Jersey and Toronto before finishing at Pittsburgh.
NOTES: The Islanders finished 13-19-9 at home. . Ottawa was without defencemen Colin Greening and Jared Cowen with lower-body injuries, plus forward Bobby Ryan (sports hernia). . Defenceman Thomas Hickey is the only Islander to appear in all 79 games this season. . The Islanders are 25-10-4 when they score at least three goals and are 6-27-7 when they do not.