Both goals for the 27-year-old Manchester City loanee were set up by another former Swansea favourite, Joe Allen, with Alfie Mawson also putting through his own net as Stoke claimed a third win on the bounce.
"It is crazy that my first goals come against my ex-team, it is disappointing for them but that is how football happens," Bony told Sky Sports.
"I have missed some chances but I knew the goals would come. Every game I push myself."
"There is no easy solution to getting confidence back," Swansea manager Bob Bradley told Sky Sports.
"When you are going through a bad stretch you have to have belief.
"It was 1-1 at half-time but we didn't start the second half in a way that suggested we could win the game," added the former United States coach, who has picked up just one point from three games in charge after replacing Francesco Guidolin.
It took less than three minutes for Bony to make his mark, pouncing to prod home from inside the six-yard box after Allen's mishit half-volley for his first goal in 23 matches for club and country.
However, the Swans levelled within five minutes as Routledge did brilliantly to head home Gylfi Sigurdsson's superb cross.
The hosts were impressive going forward and Scotland international Charlie Adam -- making his first Premier League start of the season -- hit the post with a fantastic curling effort.
Adam then conjured up another splendid effort but was this time denied by the other post.
The woodwork was proving to be a very effective 12th man for the visitors as it once gain came to the rescue after Allen's astute pass found Marko Arnautovic, who rounded goalkeeper Lukasz Fabianski only to see his effort cannon back off an upright.
Bony blazed over after being teed up by Arnautovic at the start of the second half but 19-year-old Sobhi was the creator as the hosts restored their advantage in the 55th minute.
It was another brilliant Stoke move which produced the killer third goal 17 minutes from time.
- Rooney ban -
==============
West Ham's cause was not helped by the fact they lost both wing-backs, Aaron Cresswell and Michail Antonio to injury, and in the 62nd minute they found themselves 3-1 down.
Ibrahimovic was the architect with a fine pass down the inside-right channel for Antonio Valencia, whose cross was tapped in by Martial.
The only source of regret for United was a booking for Rooney that rules him out of Sunday's trip to Everton, nixing his hopes of scoring a club record-equalling 249th goal against his old side.
Arsenal manager Arsene Wenger had been hoping to win the competition for the first time, but instead his decision to make 10 changes backfired as Southampton ended his side's 19-game unbeaten run.
Dutch midfielder Jordy Clasie's blistering drive put Southampton ahead in the 13th minute.
England left-back Ryan Bertrand added a second seven minutes before half-time, gathering Sofiane Boufal's pass and rifling a shot into the bottom corner.
"We played a very good game and to qualify for the next round is very important. We now get to play in a semi-final and that's great for the players."
Mohamed Elneny was the only player to have kept his place from Arsenal's 3-1 win over Bournemouth on Sunday, but he succumbed to illness before half-time and was replaced by Granit Xhaka.
It was Arsenal's first defeat since a 4-3 loss at home to Liverpool on the opening weekend of the Premier League season.
"We were weak in some departments and we paid for that.
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