Jean-Gabriel Pageau, Brady Tkachuk and Tyler Ennis also scored for the Senators. Craig Anderson stopped 36 shots.
With the Senators in a youth movement, the 31-year-old Anisimov has had to wait for his opportunities.
''I thought he was patient, I thought he passed the puck, he broke out well and had some offensive chances,'' Ottawa coach D.J. Smith said. ''We haven't used him much as we continue to develop the young kids. In overtime I just kind of wanted to use everybody. Everyone kind of got a shot. Brady made a good play and he ended it.''
Tkachuk finished the night with a goal and two assists, and in just his second year has already taken on a leading role with the team.
''He is really hard working and never gives up on the play,'' Anisimov said. ''He does all the stuff. He blocks shots, he hits people, goes to the net and can make plays around. He plays really well.''
John Klingberg, Stephen Johns and Joe Pavelski scored for the Stars, and Anton Khudobin finished with 42 saves. Dallas finished a three-game road trip at 2-0-1.
''There's no bad points in this league,'' Stars interim coach Rick Bowness said. ''We knew going into this month that seven of our first nine were on the road. We haven't lost one in regulation so we'll take it, but you know me, we're never satisfied. We just want to keep pushing for those points.''
Both teams were playing their third game in four nights. Ottawa was coming off a 4-2 loss to Toronto on Saturday, while the Stars beat the Canadiens 4-3 in overtime in Montreal the same night.
''I thought we were pretty solid from start to finish,'' Anderson said. ''There was no lulls in our game. We were 20 guys pulling on the rope at the same time. We were playing fast, playing hard, possessing the puck when we had to. I don't think we punted the puck too many times. I thought for sure we were playing with more poise and more confidence.''
Ennis broke a 2-2 tie midway through the third as he picked up a rolling puck and backhanded it into an open net.
The Stars tied it with 5:11 remaining when Tyler Seguin found a trailing Pavelski to beat Anderson stick side.
Seguin finished the night with two assists to extend his point streak to eight games with three goals and six assists in the stretch.
''He's flying,'' said Bowness. ''When he's playing like that you need to get him out there as much as you can.''
The Senators outshot the Stars 19-12 in a scoreless second period after each team scored twice in the first.
The Stars got lucky on the opening goal as Klingberg's shot deflected off Nikita Zaitsev's skate and right past Anderson at 3:03.
Ottawa tied it on a power-play goal by Pageau as he buried a one-timer off a great pass by Tkachuk at 7:36.
The Senators took the lead late in the period as Anthony Duclair picked up his own rebound and fed Tkachuk, who had a wide-open net with 1:50 left.
The Stars tied it 33 seconds later when Seguin won the draw and Johns' slap shot made its way through from the blue line.
NOTES: Duclair extended his goal-scoring drought to 21 games. ... Ottawa's Nick Paul played in his 100th career game. ... Dallas D Roman Polak was back in the lineup after being a healthy scratch Saturday night.
Stars: Host Arizona on Wednesday night to open a three-game homestand.
Senators: Host Buffalo on Tuesday night in the fourth of a six-game homestand.
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