The Gunners wasted a host of chances on the way to the narrow third-round replay win, with Theo Walcott and Olivier Giroud both guilty of letting opportunities go begging as the hosts struggled to turn their dominance into goals.
But influential midfielder Wilshere crowned a fine individual display in the 86th minute when he drove home a firm strike from just outside the box to send Arsene Wenger's men through to the next round, where they will travel to Brighton, and keep alive hopes of a first trophy since 2005.
A chance came early on when Walcott darted down the right and cut the ball back across goal for Giroud, who headed wide and then lashed an effort into the sidenetting.
Swansea looked to strike on the counter, with Gunners captain Thomas Vermaelen having to make a timely block as Nathan Dyer broke into the area.
Walcott was away again on 20 minutes, skipping past Chico only to be tripped by the defender as he fell, then Swansea failed to clear their lines from the free-kick and Santi Cazorla's low snapshot was well held by Vorm.
The visitors almost snatched the lead after 26 minutes when Wayne Routledge whipped the ball in from the left and former Gunner Kyle Bartley got above the static Arsenal defence to crash a header off the crossbar.
It then needed a saving tackle from Vermaelen to prevent Dyer getting his shot away after another simple pass cut through the host's rearguard, but Arsenal finally kicked into gear again as Wilshere burst into the box with the loose ball falling to Walcott, but Dwight Tiendalli was across quickly to make a smart block.
In first-half stoppage time, Walcott's corner rebounded back to Vermaelen in the six-yard box and his shot was booted off the line by Vorm.
Arsenal started the second half with intent and felt they should have been awarded a penalty after Walcott raced through onto Abou Diaby's pass. The England forward got ahead of Chico and looked to have been nudged in the back as he shaped to shoot, dragging his shot wide. However, referee Mark Clattenburg was unimpressed and pointed for a goal-kick.
Wilshere's shot was then blocked by Vorm and Walcott's rebound was kept out by Graham on the line with the Arsenal striker claiming handball, but this time Clattenburg called it right. Arsenal saw another goalbound effort, this time from Giroud, headed out by Leon Britton.
Walcott was soon away again down the left but after turning Chico inside out, he could only skew his shot wide rather than curling the ball into the far corner.
Walcott failed to connect with Sagna's deep cross, which bounced back off the inside of the far post, but Vorm was eventually beaten with just four minutes to go when Giroud touched the ball off to Wilshere at the edge of the area and he lashed a left-foot shot into the back of the net.
|Arsenal||Team Statistics||Swansea City|
|0||1st Half Goals||0|
|10||Shots on Target||0|
|10||Shots off Target||5|
|46||1st Half Poss.||54|
|50.1||2nd Half Poss.||49.9|