Spezza's second goal of the night proved to be the game-winner for Ottawa (34-31-14), but the Senators' hopes of qualifying for the Stanley Cup Playoffs ended with the Columbus Blue Jackets' overtime win against the Phoenix Coyotes.
"That's a position we put ourselves in," Spezza said. "We haven't collected enough points. You can't leave it up to other teams to lose games. Everybody wins this time of year."
Frans Nielsen scored for New York and goalie Anders Nilsson stopped 28 of 31 shots. The Islanders finished with a record of 13-19-9 at home and are wrapping up a disappointing season after making the playoffs last spring.
"We had some problems here at home," coach Jack Capuano said. "Obviously, [it's] something we have to straighten out if we want to make the playoffs [next season]. I'm proud of the fact that we gave our fans that opportunity last year in this building, it was exciting. Next year we'll be back there. We just get healthy here [and] we have a good summer."
The Senators got on the board when Spezza scored his 21st goal of the season via the power play at 12:45 of the second period. With Islanders forward John Persson in the box for tripping, Spezza took a pass from Erik Karlsson and ripped a slap shot from the left circle that beat Nilsson, who was screened by Ottawa forward Chris Neil.
Nielsen tied it with his 25th goal of the season 22 seconds into the third period. With the teams at even strength, defenseman Kevin Czuczman fired a slap shot from the left point that was stopped by Anderson, but Nielsen collected the rebound in the slot and backhanded it past the Senators' goaltender to make it 1-1.
Spezza put the Senators back in front with his second goal of the night at 12:45 when his wrist shot from a sharp angle somehow managed to go off Nilsson's back and over the goal line at the far post to make it 2-1. The goal required video review, but was quickly awarded.
"It was frustrating to get scored on to start the period," Spezza said. "I thought we played a pretty good hockey game up to that point. But we responded really well and kept getting chances."
Hemsky gave Ottawa a two-goal lead 1:39 later. Milan Michalek sprung Hemsky into the Islanders' zone with the puck and he cruised past Czuczman before wristing a low shot past Nilsson for his 13th goal of the season and second in the past three games.
"I think he'd probably like to have one back there tonight," Capuano said of Nilsson. "But, I thought he made some big saves for us. He's gonna see some action the rest of the way.
"We've got the three games left [on the road]. For me, it's continue to play these guys, give them minutes, give them opportunity and then they can make the best of their summer and come back and compete. Competition's good when you get to training camp. There's some guys that obviously have to be better if they want the minutes that they're gonna want to get next year. But when we get our team back and we're ready and we play the way that we can play, I have a lot of faith in our guys in that room that we can make a push next year for sure and get in the dance."
Anderson preserved the 3-1 lead with 2:02 remaining when he made a right pad save on Casey Cizikas after Brock Nelson poked a puck loose at center ice and sent Cizikas in on a breakaway.
Kassian's empty-net goal with 11.2 seconds left rounded out the scoring. Spezza had the primary assist for his third point of the night.
"I think we played a pretty solid, 60-minute game," Anderson said. "We had some really good scoring chances [and] their goalie made some big saves that kept it tight 'til the end. I thought special teams tonight, we were excellent. We found a way to get the win. Guys were working hard, battling and doing whatever it took to win the game tonight."
The win was the 100th for Senators coach Paul MacLean.
"We've got a long way to go," MacLean said. "It does [feel good] for sure. It's a tremendous honor. I think [it's a] credit to the players and my coaching staff. But there's still a long way to go."
New York right wing Cal Clutterbuck sustained an upper-body injury during the second period and did not return.
Islanders left wing Brett Gallant made his NHL debut. Gallant, who had one goal, one assist and 245 penalty minutes in 56 games for the Bridgeport Sound Tigers in the American Hockey League, was handed seven minutes in penalties in 6:02 of ice time.
Material from team media was used in this report