The move paid dividends as the Canucks scored three goals in the period and Mike Santorelli provided the overtime winner in a 5-4 victory over the Flames.
It was the second win for Vancouver (2-1) in as many nights following a 6-2 win over the Edmonton Oilers in their home-opener Saturday.
Santorelli deflected a pass from Kevin Bieksa to score at 3:17 in overtime.
Trailing 3-1, the Canucks kicked off their third-period comeback with Daniel Sedin assisting on a Jannick Hansen goal at 5:01, followed by Henrik Sedin setting up Santorelli's first goal of the game at 9:27.
"I know the focus is on the twins, but we split up all the lines," Tortorella said. "We had six body checks and four scoring chances in two periods, so we were dead in the water. It's an easy decision to change things up."
Dale Weise put the visitors up 4-3 at 16:06, but Calgary's David Jones tied the game again with 19 seconds remaining in regulation.
Canucks backup goaltender Eddie Lack made 32 saves in the first NHL regular-season game of his career. Roberto Luongo had made 21 saves against the Oilers.
"I was nervous on the bus ride over here, but as soon as I got into my normal warmup routine and got on the ice, the guys have been really supportive, then it felt like a normal game," said Lack, a 25-year-old Swede.
"I know coming in here there was a lot of question marks about me because I haven't played a lot. It's important for me to help the team get some points here in the beginning."
Mikael Backlund, Sean Monahan and Calgary captain Mark Giordano also scored for the Flames (1-0-2). It was the second goal in three games for 18-year-old Monahan, who was selected sixth overall by Calgary in the 2013 NHL draft.
MacDonald stopped 23 shots in his second straight start after picking up the win in Calgary's 4-2 decision over Columbus on Friday.
Tortorella lauded Lack's play and that of 24-year-old defenceman Ryan Stanton, who was claimed off waivers a few days ago.
Calgary outshot Vancouver 26-14 and out-hit them 14-6 in the first two periods.
"I think the guys decided to play for Eddie," Tortorella said. "He gave them a chance to be there and still have an opportunity to win a hockey game where really, we should have been spanked.
"It's a good test for our team to win a game like that. Those are some games you're going to have to win during the year. You're not always going to be perfect and we found a way to win."
The Flames have earned four out of a possible six points in their first three games of this NHL season, but have squandered leads in two of those games.
"We kind of sat back, we turned the puck over a couple times in the third period and with a team like that, they have a lot of talent," MacDonald said. "The Sedins, that's what they thrive on, they want to get these turnovers and they usually make you pay."
With MacDonald pulled for an extra attacker, Jones sent the game into overtime by shovelling the puck in during a goal-mouth scramble at 19:41 of the third. Officials ruled the goal valid after review.
Vancouver's Brad Richardson fed Weise with a wrap-around pass to put the Canucks up 4-3 at 16:06.
Henrik Sedin pounced on a neutral zone turnover from Calgary and dished to Santorelli who beat MacDonald with a wrist shot to tie the game 3-3.
Daniel Sedin corralled a Curtis Glencross giveaway behind the goal-line and fed it to Hansen in the slot to pull the Canucks within a goal.
Monahan, from Brampton, Ont., gave Calgary a 3-1 lead at 1:47 of the third. On an odd-man rush, he gained an outside lane, feinted a pass and slipped the puck under Lack.
Backlund scored on a breakaway off a T.J. Brodie stretch pass at 11:26 of the second period. He deked Lack and shovelled a backhand by the Vancouver goaltender's outstretched pad.
Giordano scored his first goal of the season and team-leading fourth point on a long wrist shot which beat Lack top shelf at 14:32 of the first.
Booth earned his 200th career point deflecting a Jason Garrison shot from the blue-line low past MacDonald's stick at 4:47 of the first for a 1-0 Vancouver lead.
Calgary hosts the Montreal Canadiens on Wednesday. The Canucks are at home to the New Jersey Devils on Tuesday.
Notes: Singer Corb Lund sang "Blood, Sweat and Water" during the pre-game ceremony in honour of Calgarians' efforts after the June floods that destroyed the lower level of the Saddledome . . . Calgary fell to 18-15-7 in home-openers . . . Flames forwards Mike Cammalleri (wrist) and Matt Stajan (leg contusion) and defencemen Chris Breen (shoulder) were Calgary's scratches . . . Forward Alex Burrows was out of Vancouver's lineup.Suggest a correction