Wilshere fired into Michel Vorm's net with a fine left-foot strike from outside the area after 86 minutes following relentless Arsenal pressure in the second half.
The London club's strike duo of Theo Walcott and Olivier Giroud both had shots blocked on the line as the home team mounted attack after attack.
Arsenal will travel to Championship (second division) Brighton and Hove Albion in the fourth round on the weekend of Jan. 26-27.
Premier League leaders Manchester United were playing West Ham United in another third-round replay later on Wednesday.
(Writing by Tom Pilcher, Editing by Ed Osmond)