Ryan, acquired in an off-season trade with the Anaheim Ducks, scored 26 seconds into the second period to give Ottawa a 2-0 lead.
Newly-minted captain Jason Spezza added an assist and left-winger Milan Michalek scored and added an assist. Defenceman Erik Karlsson had a pair of assists and Buddy Robinson scored Ottawa's first goal early in the opening period.
Matt Halischuk had Winnipeg's lone goal.
Robin Lehner was in goal for the Sens for the first two periods and stopped 17 of 18 shots. He was replaced by Andrew Hammond to start the third. Hammond stopped all 10 shots he faced.
Michael Hutchinson played the full game in net for the Jets and made 29 saves.
Winnipeg lost its first exhibition game Saturday night in a 4-3 shootout against Washington in Belleville, Ont.
Hutchinson let in the game's first goal when Ottawa defenceman Troy Rutkowski's shot from the point through traffic was deflected into the net by Robinson at 7:58 of the first period.
Ryan's goal was scored 26 seconds into the second period when his high shot went past Hutchinson to make it 2-0.
Winnipeg started the game without its usual top-line trio of Andrew Ladd, Bryan Little and Blake Wheeler.
Little and Wheeler skated with newcomer Michael Frolik, acquired from the Chicago Blackhawks for third- and fifth-round picks in this year's NHL Entry Draft.
Ladd, however, rejoined his linemates late in the opening period.
The Jets were sitting a number of its players who had suited up in Belleville, including forwards Evander Kane, Devin Setoguchi and Mark Scheifele.
Winnipeg took advantage of a blocked shot when its rebound was picked up by Halischuk, who went in alone on Lehner and put the puck between his glove and the post at 12:19 of the second.
The Senators gave Hammond some breathing room almost immediately.
Sixty-five seconds into the final period, Karlsson's shot from the point was redirected past Hutchinson by Michalek to make it 3-1.