Blake Coleman had a goal and an assist, and Erik Gudbranson and Andrew Mangiapane also scored to help Calgary bounce back from a 7-1 loss at Vancouver on Thursday night that snapped a 10-game winning streak.
The Flames have won 11 straight at home, tying the franchise record set in the 2015-16 season.
''We stuck with it, tried to stick to the game plan,'' Flames coach Darryl Sutter said.
Jacob Markstrom stopped 22 shots to improve to 24-11-5 this season.
Calgary fell behind less than two minutes into the game, then scored three in the first period and two in the second to take a 5-1 lead after 40 minutes.
''We just wanted to come out and have a good start,'' Tkachuk said. ''Not the most ideal start giving up one in the first couple of minutes. Bounced back, ended up winning the first period, ended up winning the second period. When you win the first two periods, it sets you up for the third.''
Frederick Gaudreau, Nick Bjugstad and Kirill Kaprizov scored for Minnesota, which has lost three straight and five of six. Kaapo Kahkonen finished with 28 saves.
''We were embarrassed tonight,'' Wild coach Dean Evason said. ''Everybody, all of us, were embarrassed tonight. There wasn't one area of our game that was good. ... They just outplayed us, outclassed us in every area of the hockey game.''
The Wild scored twice in the third to pull within two. Bjugstad skated in and scored off a pass from Matt Boldy on a rush with 6:35 remaining and Kaprizov beat Markstrom from the left circle just 70 seconds later.
Tkachuk, playing in his 400th career game, got his second of the night, an empty-netter from beyond the blue line, to push Calgary's lead to 6-3 with 1:54 left. Toffoli added another empty-netter in the final minute.
''Obviously we let up there in the third a little bit,'' Toffoli said. ''But we responded well and that's a big win for us.''
Gaudreau gave the Wild a 1-0 lead on a tip-in at 1:36 of the first period.
Calgary then scored five straight goals to take control. Toffoli tied it from the right circle with 9 1/2 minutes to go in the opening period, Tkachuk put the Flames ahead for good on a power-play goal 1:20 later, and Gudbranson's one-timer from the point made it 3-1 with 2:18 remaining in the period.
Mangiapane scored off a rebound from the left side for his team-leading 27th of the season with 3:36 left in the second.
Coleman's short-handed goal on a breakaway with 51 seconds remaining in the period pushed the lead to 5-1.
The teams complete their home-and-home set Tuesday night at Minnesota.
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