Mark Letestu added a short-handed goal, and James Wisniewski also scored for the Blue Jackets before leaving with an injury after he struck the end boards with his back early in the second period. Fedor Tyutin had two assists.
Mason, who earned his first win of the season, overcame an early goal by Johan Franzen. Damien Brunner added his fourth goal in the final seconds.
Anisimov was acquired last July in the blockbuster deal that sent captain Rick Nash to the New York Rangers. The Blue Jackets also sent a minor-league defenseman and a conditional third-round pick to the Rangers for Anisimov, Brandon Dubinsky, Tim Erixon, and a first-round pick.
Anisimov leads the offense-starved Blue Jackets with four goals.
The Blue Jackets started the second period on the power play, but they were at their best 5 on 5.
Early in the period, Columbus lost Wisniewski. He was chasing the puck when Detroit's Justin Abdelkader made contact with Nick Foligno, who then bumped into Wisniewski. He spun and hit the end boards hard, back first.
Wisniewski lay face down on the ice for a couple of minutes before medical personnel helped him turn over and eventually sit up. He was helped off the ice, skating between two teammates.
Anisimov got the first goal when Detroit defenseman Nicklas Kronwall tossed a blind pass that Anisimov intercepted. He carried the puck to the right corner, where he eluded Jonathan Ericksson, and then lifted the puck over goalie Jimmy Howard's shoulder.
Late in the period, Columbus defenseman John Moore slid a pass to Anisimov as he skated in from the left wing. Anisimov faked and got Howard to commit, then pulled the puck back and backhanded it into the vacant net for his fourth goal of the year.
It was Anisimov's first two-goal game since Nov. 11, 2010, when he was playing for the Rangers against Buffalo.
He made a run at a hat trick early in the third period, chipping the puck past Kronwall and then swooping in on Howard, who stopped his backhander.
Letestu made it 4-1 with a short-handed goal with 6:35 left, taking a lead pass from Dubinsky and sliding the puck between Howard's pads for his third of the season.
Detroit got on the board first when Franzen scored his second of the year on a rebound.