The Canucks scored three straight goals in the third period to gain a lead, but Calgary's David Jones sent the game into extra time with 19 seconds remaining in regulation as Flames goaltender Joey MacDonald was on the bench for a sixth attacker.
Dale Weise, Jannik Hansen and Santorelli scored in the third after the Canucks fell behind 3-1 early in the period. David Booth also scored for the visitors in the first period.
It was the second win in as many nights for Vancouver (2-1-0), who beat the Edmonton Oilers 4-2 on Saturday in Vancouver's home-opener.
Canucks backup goaltender Eddie Lack made 32 saves in his first career regular-season NHL game. Roberto Luongo had made 21 saves against the Oilers.
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.
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.
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 to tie the game. Jones shook off a check along the back boards to feed Calgary's new captain out front.
Backlund scored on a breakaway on a T.J. Brodie stretch pass at 11:26. Backlund deked Lack and shovelled a backhand by the Vancouver goaltender's outstretched pad to put Calgary up 2-1 and Monahan, who's from Brampton, Ont., extended Calgary's lead at 1:47 of the third. He gained an outside lane in the offensive zone, took a pass and slipped the puck under Lack.
Hansen scored at 5:01 to cut into Calgary's lead and Canucks captain Henrik Sedin dished to Santorelli at the top of the offensive zone to tie the game 3-3. Daniel Sedin corralled a Curtis Glencross giveaway behind the goal-line and dished it out front to Hansen in the slot while Santorelli fooled MacDonald with a wrist shot.
Brad Richardson fed Weise with a wrap-around pass to put the Canucks ahead by a goal at 16:06.
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.