In his first start, Jacob Markstrom made 12 saves in 40 minutes of action before giving way to Dustin Wolf, who made nine stops in the final period to complete the shut out.
MacKenzie Weegar and Jonathan Huberdeau, with an empty netter, also scored for Calgary, which improved to 3-1-0 through the halfway point of its pre-season schedule. Nazem Kadri recorded one assist in his Flames debut. The Oilers fell to 1-2-0.
The Flames surged in front 2-0 in the second period with Sutter, son of head coach Darryl Sutter, opening the scoring on a rebound at 6:56. At 16:36, Stone ripped a slap shot past Stewart Skinner to double the lead.
Stone, in camp on a professional tryout, has been with the Flames since being acquired from the Arizona Coyotes at the trade deadline in 2016-17. Since then, he signed a three-year deal that he was eventually bought out from, only to remain in Calgary, signing three consecutive one-year deals.
The 32-year-old is competing for a depth role on the blue line where six jobs are already locked up with newcomer Weegar, acquired from the Florida Panthers, Chris Tanev, Rasmus Andersson, Noah Hanifin, Nikita Zadorov, and Oliver Kylington.
Stone dressed for 11 regular-season games last season, but shone in the playoffs, picking up five points (two goals, three assists) in nine games.
Kadri, general manager Brad Treliving's prize free agent addition in the off-season after signing a seven-year, US$49-million deal, centred a line with Mikael Backlund and Blake Coleman.
Kadri made his presence felt in the first period when he got out on the Flames' No. 1 power play with Huberdeau and the new-look unit hemmed the puck in the Oilers end for nearly the entire two minutes. However, they couldn't solve Edmonton starter Calvin Pickard.
Pickard finished with 16 saves on 17 shots. In relief, Skinner allowed two goals on 16 shots.
Most of the Oilers' top-end regulars did not make the trip, with Connor McDavid, Leon Draisaitl, Ryan Nugent-Hopkins, and Evander Kane among the absent forwards. Darnell Nurse was missing from the blue line and newly signed Jack Campbell also did not make the trip.
Notes: The Flames, who have the day off Thursday, still have 58 players in camp _ 33 forwards, 19 defencemen, and six goaltenders.