Scott Sabourin also scored in regulation for the Senators, who earned their first shootout win of the season. Marcus Hogberg made 25 saves for his fifth win.
Anisimov's goals won't count as a hat trick, but the forward is labeling it one.
''The puck crossed three times the goal line,'' Anisimov said with a laugh.
It was a career night for the 31-year-old, who had 10 shots on goal.
''I always want to shoot the puck in games, but sometimes you see better opportunity and you pass, but the pass (is not) connected and it's a turnover,'' Anisimov said. ''I decided why not shoot the puck and the puck was around me always and I just needed to shoot it.''
Frans Nielsen, Dylan Larkin and Sam Gagner scored for the Red Wings, who have lost nine of 10. Jonathan Bernier stopped 36 shots.
''We took a step forward,'' Red Wings coach Jeff Blashill said. ''We were better in certain areas than we've been. We can still be better, for sure, but we took some steps forward. We need to win, so it was disappointing not to get the win, but it was certainly a better effort than the last couple nights.''
The Red Wings were coming off an embarrassing 7-1 loss to Minnesota on Thursday and had talked about giving a better effort.
''As a group we felt ashamed a little bit,'' Jonathan Bernier said. ''We're professionals, obviously it's a tough season, but you've just got to compete and come in every night and give your best and that's all you can do. Obviously we put ourselves in this position, but you've just got to compete until the end.''
The Senators tied the game 2-2 just 1:48 into the third period as Sabourin scored his first since the season opener.
Anisimov gave Ottawa the lead with his second goal of the game when he took a pass and scored over Bernier.
The Red Wings came right back and tied the game with a power-play goal. Gagner tipped Filip Hronek's point shot and handcuffed Hogberg on the redirection.
Detroit extended its lead to 2-0 early in the period as Tyler Bertuzzi dropped a pass to Larkin, who fired a wrist shot and beat Hogberg to the glove side.
The Senators cut the lead in half with a power-play goal late in the period. With Nick Paul screening Bernier, Anisomov got it through a crowd for his 14th of the season.
The Red Wings took a 1-0 lead four minutes into the first period on Nielsen's fourth of the year.
The Senators had a number of chances on the power play, but Bernier came up with a couple of big saves.
Shortly after, Adam Erne pushed his way to the net, but the puck trickled off his stick to Nielsen, who put it behind Hogberg.
NOTES: Ottawa's Thomas Chabot played his 200th NHL career game. ... Colin White and Anthony Duclair missed their third straight game due to injury. ... This was the final meeting between the teams. Ottawa won the season series 2-0-1.
Red Wings: Host Colorado on Monday.
Senators: At Piitsburgh on Tuesday.
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