With 9:30 minutes left, Foligno took a wrist shot from just above the circle and then put his rebound past diving goalie Brian Boucher, who had smothered another drive by Foligno a few minutes earlier.
Winchester sealed the victory with a short-handed, empty-net goal in the final minute.
It was the first shutout this season for the Senators, who have two wins in their past six games.
The Flyers were blanked for the third time in six games. They are 1-3 under new coach Peter Laviolette, who replaced the fired John Stevens a week ago after Philadelphia's two consecutive shutout losses.
Philadelphia, which lost six of seven before the coaching change, has scored only 11 goals in seven games. Six of those came in a 6-2 victory over the New York Islanders on Tuesday.
The Senators scored 12 goals in their previous five games, earning their lone win in a shootout against Anaheim.
Philadelphia's best scoring opportunity came with 5:47 in the second period, but Elliot stretched across the net to thwart Scott Hartnell's backhanded breakaway attempt.
NOTES: Jeff Carter played in his 243rd consecutive game, the fifth-longest streak in Flyers history. ... Elliott posted his other NHL shutout on March 29 against Tampa Bay. ... Boucher is 1-5-1 in eight career starts against Ottawa.