The Gunners piled on the pressure during the final 20 minutes of the match and finally got their reward when the England midfielder fired in from the edge of the area.
Olivier Giroud, Theo Walcott and Thomas Vermaelen also went close only to be denied by Swansea keeper Michel Vorm.
Kyle Bartley went closest for the Swans when his header struck the bar.
His was one of four good opportunities created by Michael Laudrup’s men in the opening 45 minutes as they dominated possession. The second half was a very different story as Arsenal finally got into their stride.
Wilshere was key to the Gunners improvement after the break, pushing higher up the field having had to support his fellow midfielders as they attempted to put the shackles on the sprightly Nathan Dyer in the first half.
The England midfielder could have scored another had Vorm not got down well to prevent his drive from going in. In fact the Dutch keeper was brilliant all night, blocking Vermaelen’s shot before the break and then repeating the trick to keep out efforts from Giroud and Walcott.
Walcott also had a volley which struck the chest of striker Danny Graham, while Bacary Sagna’s intended cross struck the Vorm’s far post.
But with five minutes remaining, the last line of defence was breached when Giroud flicked Santi Cazorla’s pass into the path of Wilshere, who rammed in his shot from the edge of the area – his first goal in the FA Cup.
It was the least the Gunners deserved, but few would have guessed they could produce such a performance after a tepid display in the first half.
The home side struggled like manager Arsene Wenger did with the zip of his coat, and were lucky not to go behind when Graham fired wide after he was put through by Kemy Agustien.
Bartley then hit the post with his header while Dyer, who faded in the second half, saw two efforts blocked by Vermaelen.
Laudrup brought on Michu midway through the second half, but the club’s top scorer found himself helping out the defence on a night when Arsenal showed their bite.
The Gunners will now play Championship side Brighton, who have already disposed of one Premier League side, Newcastle.