Toronto dropped its third straight since the Winter Olympics. The Maple Leafs had won seven consecutive at home, their best stretch there since a nine-game run in the 2006-07 season.
"We had a mediocre second period and it cost us the game," Toronto defenseman Cody Franson said. "We got away from our game plan. We tried to get cute with the puck and turned it over too many times.
"When you play a team that has that kind of transitional ability, that's not what you want to do."
Prout, a defenseman, and Anisimov scored right after each other. Sergei Bobrovsky made 28 saves for the Blue Jackets, who completed a three-game season sweep of Toronto.
Mason Raymond had the lone goal for Toronto, bringing the Maple Leafs within one with 4:15 remaining. Toronto made a late flurry after goalie James Reimer was pulled for an extra skater, but couldn't tie it.
Reimer made 31 saves.
The Leafs carried the play for the majority of the first period and skated with desperation in the third, but couldn't break through the Blue Jackets' defense.
"For the first period we had lots of life and we were winning battles," coach Randy Carlyle said. "In the second period we just started to stall in the neutral zone. We're a team that needs to push the puck up the ice and establish a forecheck, and we didn't do that for the first 13, 14 minutes of the second period."
Toronto had won five of six before the Olympic break to climb up the Eastern Conference standings, but the Maple Leafs are 0-1-2 since the NHL resumed play last week. The Maple Leafs are fifth in the conference, but five teams are within five points.
"We were rolling pretty good there and feeling quite confident in what we were doing," Franson said. "When you're rolling like that, sometimes it can be difficult to find that smoothness in how you were performing before."
Toronto dropped overtime games to the New York Islanders on Thursday and at Montreal on Saturday
"The three losses hurt because we had points available to us on the road," Carlyle said. "That's the hard part, losing two overtime games on the road when we had one-goal leads in both games."
Reimer, making his first start since Jan. 25, was sharp early as the Blue Jackets applied pressure, including a clear-cut breakaway by Nick Foligno less than four minutes into the game that was denied.
Aided by a pair of power plays in the middle of the opening period, Toronto took over the momentum and had a handful of scoring chances but couldn't solve Bobrovsky.
Prout opened the scoring when his slap shot from the left point beat a screened Reimer at 9:44 of the second. It was the defenseman's first goal in 30 games this season and his second in 63 career NHL games.
"He's one of those guys that you root for and his teammates root for," Blue Jackets coach Todd Richards said.
A few minutes later, Anisimov gave Columbus a two-goal lead when he converted Nathan Horton's pass at the side of the crease at 12:38.
Columbus went 0 for 1 on the power play. Toronto was 0 for 2.
The Maple Leafs will visit the New York Rangers on Wednesday, play host to the Philadelphia Flyers on Saturday, and then kick off a five-game road trip with three games in California next week.
The Blue Jackets routed the Maple Leafs 6-0 in Toronto in the previous meeting between the teams on Nov. 25. ... Toronto scratched fourth-line forwards Colton Orr and Frazer McLaren. Columbus sat wingers Blake Comeau and Corey Tropp and defenseman Cody Goloubef. ... Ryan Johansen leads the Blue Jackets with 24 goals and 47 points in 61 games. ... Columbus defenseman Ryan Murray left the game in the first period because of a lower-body injury and didn't return.