Nick Ritchie and Ondrej Kase scored in regulation for Anaheim, which has won four of its last five.
John Gibson made 30 saves as the Ducks improved to 2-0-0 on their five-game road that started with a 3-1 win in Los Angeles against the Kings.
''It's pond hockey a little bit when it's three-on-three and could go either way,'' Ottawa's Jean-Gabriel Pageau said of the overtime period. ''We had our chances, they had their chances and it goes into a shootout and that's not hockey, but that's what it is right now so not much we can do.''
Drake Batherson and Artem Anisimov scored for Ottawa. Marcus Hogberg stopped 26 shots.
''Really I thought we had a good game,'' Senators coach D.J. Smith said. ''We had our chances to win in overtime and we didn't and they win in a shootout.''
With 10 home games this month, the Senators hope to improve their record at Canadian Tire Centre, where they're 1-4-5 in their last 10.
Anaheim coach Dallas Eakins thought his team played slow and turned the puck over far too many times and will need to control the pace better if they hope to continue to find success on this road trip.
''We were very lucky,'' said Eakins. ''I'm not sure we got what we deserved. That was not a very good game by us, but John Gibson held us in and we got some timely scoring.''
Anaheim opened the scoring midway through the first period when Ritchie tipped Hampus Lindholm's shot to beat Hogberg.
The Senators tied it early in the period as Batherson scored a power-play goal for his second of the season.
''I'm definitely feeling a lot more comfortable than I was at the start of the year,'' Batherson said. ''I think if you ask any guy they're the same way. The more time they're here the more they get comfortable and that's how I'm feeling. I had some space to skate with the puck and that's what I tried to do.''
Kase took a bounce off the back boards and tucked in a backhand to make it 2-1 at the 13:35 mark of the second period.
Anisimov's 10th of the season tied the game 2-all with a power-play goal early in the third, tipping Colin White's shot past Gibson.
Ottawa appeared to tie the game late in the period, but the Ducks challenged and, upon review, was deemed offside.
NOTES: Anthony Duclair took the ceremonial faceoff for the Senators as Willie O'Ree dropped the puck as part of the NHL's Black History Month celebrations. ... Cody Goloubef and Mikkel Boedker were healthy scratches for the Senators.
Ducks: At Montreal Canadiens on Thursday.
Senators: Host Colorado Avalanche on Thursday.
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