The England midfielder - out for more than a year with ankle and knee injuries before returning to action this season - capped a man-of-the-match display to finally beat inspired Swansea goalkeeper Michel Vorm with a 20-yard strike in the 86th minute to set up a trip to Brighton and keep alive hopes of Arsenal's first trophy since 2005.
The Gunners were much improved from the lacklustre 2-0 Barclays Premier League defeat to Manchester City on Sunday and, but for a combination of poor finishing and brilliant defending, would have had this replay wrapped up well before Wilshere's fine late effort.
With France striker Olivier Giroud returned to the starting line-up and England forward Theo Walcott switched back to the right flank, Arsenal opened up full of the kind of movement which had been lacking against City.
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.
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.
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.
Swansea - who won the first leg of their Capital One Cup semi-final 2-0 at Chelsea - remained dangerous on the break.
Danny Graham ran down the left and with Jonathan de Guzman clear on the other flank, could only hoist his cross too high and behind.
Slowly the Arsenal fans inside the crowd - with 58,359 given as the 'tickets sold' figure - cranked up the temperature as Giroud flashed a header across the face of goal from a corner.
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 then 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.
The visitors made a change with 20 minutes left when Michu - who scored a late brace to sink Arsenal here in the Premier League at the start of December and also netted in the 2-2 FA Cup draw at the Liberty Stadium 10 days ago - replaced Graham.
Arsenal continued to press and Wilshere's 20-yard drive was beaten away by Vorm, who then collected a scuffed effort from Giroud.
In a rare moment of action at the other end, Chico headed a corner into the side-netting.
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.
- Sports & Recreation
- Theo Walcott
- Olivier Giroud
- Michel Vorm