"There's nothing real positive about it at all," Ottawa coach Paul MacLean said. "We haven't won a game in it seems like forever. The league is hard, winning is hard to do. And we haven't played hard enough in the last five games to produce more than what we say is good opportunities.
"We need to do more. Not a whole lot more, but we need to do a little bit more."
Ottawa is sixth overall in the Eastern Conference, however, ninth-place Winnipeg is just two points behind the slumping Senators.
Stamkos snapped a 2-2 tie with a power-play goal from the near post off a pass through the slot by Vincent Lecavalier with 5:04 to play.
Ottawa, which entered with the NHL's second-best penalty-killing unit, gave up two power-play goals in seven short-handed situations.
"We're not concerned about playoff spots," Senators goalie Craig Anderson said. "We're concerned about doing things right and coming to the rink, getting better. At the end of the day, we control what we can control, and that's how we play and how we come to work. The standings, wins, they'll all take care of themselves."
Ben Bishop made 31 saves for Tampa Bay. He was traded to the Lightning from Ottawa for center Cory Conacher on April 3.
Tampa Bay also got goals from Pierre-Cedric Labrie and Lecavalier. Martin St. Louis assisted on Lecavalier's goal for his 900th point. The right wing, the 98th NHL player to reach the milestone, also assisted on Stamkos' goal.
The Lightning, 13th in the Eastern conference, trail Southeast Division-leading Washington by eight points.
"The odds are against us, and they know what's going on," Tampa Bay coach Jon Cooper said. "They have every reason to sit here and say, this isn't going to work. I don't have to go down and block a shot. I don't have to go to the net. But, they're not doing that."
Tampa Bay took a 2-1 lead early in the second on goals by Labrie and Lecavalier. After Labrie beat Anderson from in-close at 2:35 for his first NHL goal, Lecavalier gave the Lightning a one-goal advantage during a 5-on-3 power play 1:53 later.
"We're not going to quit here until you see that line across the Tampa Bay Lightning, that they're officially eliminated," Stamkos said.
Alfredsson put the Senators up 1-0 at 14:29 of the first. He swatted a waist-high puck into the net after Bishop got a piece of Turris' shot with his glove.
Turris hit the post shortly after Alfredsson scored and had a potential goal with 2:08 left in the second disallowed when it was ruled he knocked the puck into the net with a high stick. The call was upheld after a video review.
Alfredsson has 425 goals, tying him for 70th place on the NHL career list with Brian Propp. The right wing also tied Frank Mahovlich for 56th place with 1,103 points.
Tampa Bay LW Ryan Malone played after missing eight games with a left shoulder injury. ... Anderson has played in consecutive games. He missed 19 games after hurting his ankle in late February. ... Lightning D Victor Hedman missed his third straight game because of a lower body injury. ... Tampa Bay G Anders Lindback, sidelined the past eight games with an ankle injury, has resumed working out with the team. ... Ottawa D Erik Karlsson (Achilles tendon surgery) has resumed skating. ... Labrie sat out the previous six games with an upper body injury.
Copyright by STATS LLC and The Associated Press
TAMPA, FL - APRIL 9: Tyler Johnson #63 of the Tampa Bay Lightning warms up before the game against the Ottawa...