Columbus climbed into a tie for seventh with the Rangers in the Eastern Conference. The Blue Jackets have won three in a row and are vying for only their second postseason appearance since joining the league in 2000.
Columbus scored on the first shot of the game and bolted to a 3-0 first-period lead on goals by Boone Jenner, R.J. Umberger and Anisimov against backup Dan Ellis. Ellis was shaky early in making his first appearance since Feb. 1.
Sergei Bobrovsky made 31 saves, one on an open shot by Jamie Benn with 5 minutes left with the Blue Jackets clinging to a one-goal lead. Bobrovsky solidly tracked loose pucks near the crease the entire game to improve to 5-0-1 against Dallas.
Defensemen Alex Goligoski and Trevor Daley had goals early in the third period for the Stars. With the loss, Dallas, which had won five of seven, still sits in eighth place in the Western Conference.
Leading 3-0 entering the third period, Columbus seemed in control even though it had been drastically outshot since taking the big lead near the midpoint of the first. But the Stars finally broke through, scoring about two minutes apart.
Goligoski fired home a rebound on the power play from near the crease, his fourth goal and second in as many games after a 48-game drought. Daley followed with his fifth on a redirection from between the circles off soft point shot by Sergei Gonchar.
Dallas kept coming, but Anisimov made it 4-2 with his 18th and fifth in four games on a backhand after circling toward the crease from behind the net.
The Blue Jackets benefited from one stick-handling flub after another by Dallas' defensive corps to easily pull ahead in the first period.
Skating into the Dallas zone with speed, Jenner sent a long snap shot from the left circle into the far corner above Ellis' outstretched glove for his 11th.
Aaron Rome then swept the puck off Brandon Dubinsky's stick in the slot to a streaking Umberger, who scored his 17th
Just over four minutes later, the Stars' Jordie Benn cleared the puck into traffic to teammate Cody Eakin, who bobbled it into the slot to Anisimov. The big center turned and sent a backhander past Ellis to make it 3-0.
The Stars outshot Columbus 12-1 in the second period but didn't have many prime scoring chances other than two by Tyler Seguin on a power play that Bobrovsky smothered.
NOTES: The Stars traded injured D Stephane Robidas to the Ducks on Tuesday for a conditional fourth-round pick. ... Columbus played without injured defensemen Fedor Tyutin and Ryan Murray, both out with lower-body injuries. ... Dallas entered 16-5-3 against the Blue Jackets in Nationwide Arena. ... Dallas C Rich Peverley did not play due to an upper-body injury. ... Stars LW Antoine Roussel become the fourth player from France to appear in at least 100 NHL games.