Teenage midfielder Barkley, on loan from Everton, struck early in each half and Antonio steered home a third in the closing stages to lift the pressure on Wednesday manager Dave Jones, who celebrated his side's first win in 10 league matches.
Ipswich, who parted company with manager Paul Jewell on Wednesday, have yet to win a league game at Portman Road this season and their winless run has now stretched to 11 matches.
The visitors attacked from the kick-off and when Lewis Buxton's shot was blocked by Luke Hyam following Antonio's cross, Barkley rifled the ball home, low under Ipswich goalkeeper Stephen Henderson in the second minute.
Ipswich defender Luke Chambers headed his side's first chance straight at Wednesday goalkeeper Chris Kirkland from Josh Carson's corner, while Hyam dragged his shot from the edge of the area wide.
Wednesday went close to doubling their lead in the 16th minute when Antonio's goalbound header from Jay Bothroyd's cross was blocked by Ipswich defender Danny Higginbotham, but Ipswich were not without their chances.
Aaron Cresswell's 20-yard free-kick flew over the crossbar, fellow midfielder Richie Wellens fired a 25-yard shot just over and Higginbotham headed Lee Martin's free-kick narrowly wide in the 40th minute.
Ipswich stepped up their pursuit of an equaliser before the break as Higginbotham headed Cresswell's free-kick over the crossbar and Carson's stinging drive was well held by Kirkland.
Barkley and Ipswich striker DJ Campbell traded shots from outside the area after the restart before the visitors struck again.
Antonio headed the ball inside from the left and Barkley stepped inside on the edge of the area and arrowed a low 20-yard shot into Henderson's bottom right-hand corner.
Hutchings sent on striker Michael Chopra for Daryl Murphy in the 60th minute, but Antonio went close to adding a third for the visitors with a stinging shot held by Henderson.
The Owls continued to look the more likely to add to the scoreline and the third goal arrived in the 81st minute.
Striker Chris O'Grady played the ball across the six-yard box and Antonio was left to convert the simplest of finishes from close range.