Olli Jokinen, Bryan Little and Mark Scheifele all notched a goal and an assist as the Winnipeg Jets seemed to be clicking with Maurice as their leader.
Maurice was named the Jets' bench boss on Sunday after the team fired Claude Noel.
Other Jets who recorded multi-points in Thursday's win were Andrew Ladd, Chris Thorburn and Blake Wheeler, who all notched a pair of helpers.
The Jets got off to an impressive 4-0 lead as Ondrej Pavelec only had to make 11 saves over the first 40 minutes. The Flames battled back in the third with goals from Mark Giordano and Lance Bouma, but it was far from enough.
With the win, Winnipeg improves to 21-23-5, while Calgary drops to 16-26-6. The Flames will next play in Vancouver on Saturday.