Namestnikov, who was acquired in a trade with the New York Rangers on Monday, made it 3-2 with 3:03 remaining in the third period when he finished 2-on-1 with Connor Brown. He scored into an empty net at 19:38 for the 4-2 final.
"Never fear an opponent," said Namestnikov, who played for the Lightning from 2014-18. "You've got to go out and play your game, not think about who's out there. I think that's what we did today. We limited their chances and got rewarded."
WATCH: [All Lightning vs. Senators highlights]
Jean-Gabriel Pageau had a goal and an assist, and Craig Anderson made 19 saves for the Senators (1-3-0).
"We had a lot of guys play hard today," said Senators coach D.J. Smith, who got his first NHL win. "Right from the get go, I thought we played hard. We played hard defensively, played hard in the [offensive] zone. We're doing a better job trying to limit shots and we had a lot of guys really play hard, especially Pageau with the old Gordie Howe hat trick tonight. You don't see that much anymore."
Ondrej Palat and Luke Witkowski scored, and Curtis McElhinney made 30 saves for the Lightning (2-2-1), who have lost three of their past four games.
"I don't think it was for lack of effort," Tampa Bay defenseman Ryan McDonagh said. "It was really just disjointed and really not understanding the situations and momentum swings of the game. I think that's something we need to learn as a team. We go out and get in our groove and feel confident, but when it's not there, you've got to simplify and you've got to be really good in your support."
Colin White scored with a wrist shot short side from the right circle to give the Senators a 1-0 lead at 13:44 of the second period. Bobby Ryan had the secondary assist on the goal, his 300th in the NHL.