Gustavsson has two shutouts in 84 career appearances spread over three seasons. The Maple Leafs won their second straight on the heels of a three-game losing streak.
Phil Kessel, who began the night tied for second on the NHL's goal-scoring list, produced his 23rd goal of the season 5 minutes into the game.
All four goals were surrendered by Chris Mason on 25 shots.
The Jets have lost two in a row, having suffered a 7-3 defeat at the hands of Montreal Wednesday.
Celebrity Families of 2014 [PHOTOS]