Thomas Chabot, Shane Pinto and Derick Brassard also scored for the Senators, scored four straight over a 12:07 stretch after falling behind 1-0 in the first period. Magnus Hellberg stopped 30 shots in his season debut.
''I've been waiting to get a start for a couple of weeks now, so obviously it was nice to finally be in net,'' said Hellberg, who signed a contract with the Senators during the summer after spending the previous five seasons in the KHL. ''I have that inner drive to play at the best level and I feel that I can play in this league as a full-time job. Obviously I'm really happy that it worked out this summer and that I'm able to play in this league.''
Joel Kiviranta and Wyatt Johnston scored for the Stars and Scott Wedgewood finished with 28 saves. Dallas (4-1-1) lost in regulation for the first time this season.
Tkachuk skated through the left circle and got past Stars defenseman Nils Lundkvist and beat Wedgewood for a 2-1 lead at 4:17 of the third.
Pinto made it 3-1 nearly six minutes later as he scored for the fifth consecutive game, becoming the first Senators rookie to accomplish the feat.
''It's just going in for me right now,'' Pinto said. ''It's getting pretty lucky but I'm just happy we found a way to win that one. That was another mature win by us and we just have to keep it rolling.''
Brassard, also making his season debut, went down the ice and wired a shot past Wedgewood for a three-goal lead with 8 1/2 minutes remaining.
During their winning streak, the Senators have outscored opponents 10-1 in the third period.
''We understand how hard we have to work in a game to get to the third period and not be down,'' Ottawa coach DJ Smith said. ''I thought we worked really hard in the second to set that up.''
Johnston pulled the Stars back within two with 2:16 remaining.
Dallas had the first eight shots of the game and Kivirianta got the Stars on the scoreboard at 6:31 of the first period.
''I liked our start,'' Stars coach Peter DeBoer said. ''I thought we came out of the gate really good. I thought as the game wore on, their commitment and energy increased and ours decreased, so we'll have to look at why. ... But you have to give them credit. They were the hungrier team for most of the game tonight. Tough to win in this league when that's the case.''
Chabot tied it with 37 seconds remaining in the second as he ripped a point shot past Wedgewood while the Senators were on their third power play of the period.
With starter Anton Forsberg sidelined, Kevin Mandolese was the backup goaltender for the Senators as he was called up from Belleville of the AHL. ... Dallas has not lost any man games to injury so far this season.
Stars: At Boston on Tuesday night to complete a four-game trip.
Senators: Host Minnesota on Thursday night to finish a five-game homestand.
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