Gary Madine and Ryan Lowe also scored for the hosts with Steven Schumacher grabbing Bury's goal.
Antonio fired into anunguarded net in the 11th minute after Lowe knocked the ball to the side of Trevor Carson in the area, leaving the keeper stranded off his line.
Lowe had a great chance to score the second when Madine picked him out with a fine through ball but the striker's shot was saved by Carson.
Madine found the net from close range following Rob Jones' knock- down to make it 2-0 six minutes into the second half.
Seven minutes later Antonio scored again with a low drive after Mike Jones' cross found him unmarked.
Lowe put Wednesday further ahead after Antonio's pace exposed Bury's defence on the counter-attack before Schumacher pulled one back with 18 minutes left.