Ryan, acquired in an offseason trade with the Anaheim Ducks, scored 26 seconds into the second period to give Ottawa a 2-0 lead.
New captain Jason Spezza added an assist, and Milan Michalek had a goal and added. Defenseman Erik Karlsson had two assists, and Buddy Robinson scored Ottawa's first goal early in the first period.
Matt Halischuk had Winnipeg's lone goal.
Robin Lehner stopped 17 of 18 shots in two periods of action for Ottawa. He was replaced by Andrew Hammond, who stopped all 10 shots he faced.
Michael Hutchinson played the full game for the Jets and made 29 saves.
Winnipeg lost its first preseason game Saturday night, 4-3 in a shootout to Washington.