Rochdale never got to grips with the impressive Johnson and he capped a fine performance with his second goal five minutes after the break as the Owls booked their place in Wednesday's second-round draw.
Wednesday came flying out of the traps and Tommy Miller saw a shot saved by Kenny Arthur but five minutes later Marcus Tudgay flicked the ball on for Esajas who sent a looping volley from 25 yards over the despairing Rochdale keeper.
The Owls doubled their lead with 19 minutes gone as Frank Simek's long ball into the box rebounded into the path of Johnson who kept his composure and slammed the ball past Arthur from six yards.
Rochdale could not cope with Johnson and with 27 minutes gone he latched onto Tudgay's clever flick and turned superbly before firing in a shot that was blocked by Arthur.
Five minutes before the break Rochdale almost found a way back into it when Gary Jones played the ball into Chris Dagnall but his shot was saved by Wednesday goalkeeper Lee Grant.
Wednesday once again started well in the second half and another clever flick from Tudgay found Miller who just shot wide.
Moments later the Owls killed the game off as the Rochdale defence let Mark Beevers' long ball out of defence run straight through to Johnson who coolly lobbed the ball over the onrushing Arthur.
Wednesday continued to dominate with Darren Potter and substitute Francis Jeffers both going close to extending the score line as the Owls eased home with Rochdale well beaten.