Eller scored in the first period and Brendan Gallagher and P.K. Subban added power-play goals in the second for Montreal (18-5-4), which played loosely in its first game at home after a five-game road trip. Carey Price made 29 saves to get the win.
Ottawa goalie Robin Lehner was sharp in the third period as Montreal had the man advantage three times to stretch their lead in shots for the game to 45-32.
The result gave the teams two wins each in their four-game season series.
Galchenyuk scored on a deke and Eller on a drag move in the shootout, while Jakob Silfverberg had the only goal for the Senators.
Wiercioch, a rookie defenceman, scored his third goal in the last five games after not scoring in his first 26 in the NHL to tie things 3-3 early in the third.
With a goal and an assist, Subban has points on Montreal's last nine power-play goals.
The Canadiens opened the scoring as recent callup Gabriel Dumont earned his first NHL point as he worked the puck from behind the net to Eller, who lifted it over Lehner at 7:02 of the first period.
An Ottawa power play was nearly over when Silfverberg made a sweet pass from a crowd in the corner to Zibanejad, who beat Price with a quick wrist shot to tie it at 10:26.
Pacioretty skated in off the boards and slipped a shot between Lehner's pads 6:40 into the second and Subban scored from the point on a shot that hit Ottawa defenceman Chris Phillips in front at 17:21.
Time was about to expire in the second period when Alfredsson aimed a shot over Price's shoulder that hit the camera inside the net and bounced out. Video replay confirmed the goal at 19:58.
Wiercioch tied it 1:50 into the third when he moved across the blue-line, put a move on Andrei Markov and scored on a wrist shot from the slot that beat Price between the legs.
Notes: The banged-up Senators are still without Craig Anderson, Erik Karlsson, Jason Spezza, Jared Cowan and Milan Michalek. Peter Regin didn't dress. ... UFC star Georges St-Pierre got a big ovation as he dropped the puck for the ceremonial faceoff. ... The Montreal Impact's new Swiss coach, Marco Schallibaum, attended the game, although the Canadiens' Swiss players, Rafael Diaz and Yannick Weber, are both out with injuries.