Anderson drew Montreal within 4-3 at 18:48 with his second of the game.
Alex Formenton made it 5-3 at 2:55 of the third period when he redirected a shot from Anisimov, who missed the previous eight games because of an upper-body injury.
"When you're injured and you're watching games, you try to find the weak spots on the other team because we only play against Canadian teams this year," Anisimov said. "And you try to figure out what to do on the ice when you're watching the game play from upstairs, and when you're on the ice you try to do those things on the ice."
Drake Batherson scored into an empty net to make it 6-3 at 17:05.
"I thought we played hard and well," Forsberg said. "We were playing a team game out there and threw pucks at the net and scored a lot of goals, which is always nice as a goalie."
Ottawa took a 1-0 lead on Tkachuk's power-play goal 42 seconds into the first period.
Tyler Toffoli tied it at 1-1 with his Canadiens-leading 19th goal on a power play at 5:31 in his first game since March 19. He missed three games because of a lower-body injury.
Connor Brown's shorthanded goal at 12:07 put the Senators up 2-1. Anderson tied it 2-2 at 6:58.
NOTES: Senators forward Colin White did not play and will not face the Winnipeg Jets on Monday because of an upper-body injury. … Forsberg made his second start since he was claimed on waivers from the Winnipeg Jets on March 17. It was his second NHL win since 2017-18 and his first since March 7, 2020, his only one in three games with the Carolina Hurricanes last season. … The Canadiens have the day off Sunday. Eric Staal will skate at their practice facility in Brossard, Quebec. The 36-year-old forward completed a seven-day quarantine after he was acquired in a March 26 trade with the Buffalo Sabres. He is expected to take part in the morning skate Monday and make his Montreal debut against the Edmonton Oilers.