Stoke also hit the woodwork three times in the first half, twice through Charlie Adam and once by Marko Arnautovic.
Swansea defender Alfie Mawson put through his own net 10 minutes after halftime. Wayne Routledge grabbed the only goal for the visitors in the eighth minute.
Stoke's victory helped them climb five places to 12th in the table, with 12 points from 10 games. Swansea remained second from bottom, three points ahead of Sunderland.
"Our attacking play was really good today, any number of chances could have gone in," said Stoke manager Mark Hughes.
"We do not feel like a team that belongs in the bottom half.
Wilfried and Joe have been outstanding since they came here and they are doing a great job."
Bony tapped in the opening goal from close range after a corner was only half cleared and Allen delivered an inviting ball into the penalty box.
Routledge equalised with a header after pouncing on a perfect cross from the right by Swansea playmaker Gylfi Sigurdsson.
Adam was unlucky to see two drives bounce off the woodwork while Arnautovic had an effort rebound from the post after he rounded goalkeeper Lukasz Fabianski following another penetrating pass by Allen.
Stoke went 2-1 up when Mawson knocked the ball into his own net after a jinking, twisting run in the penalty area by substitute Ramadan Sobhi.
Allen was again involved in the third goal for the home team when he set up Bony to head the ball in after some trickery in the box from Arnautovic.
"We had some good stretches but overall we lack confidence and in key moments we were second best," said Swansea manager Bob Bradley.
"When you are going through a bad stretch you have to have belief. It was 1-1 at halftime but we didn't start the second half in a way that suggested we could win the game."
(Writing by Tony Jimenez; Editing by Pritha Sarkar)
