Arsenal had to come from behind after Sergio Aguero scored his 30th goal of the season for City in the 63rd minute, racing away on to a pass from Yaya Toure.
Spanish international Nacho Monreal equalised nine minutes later, from a cross by Arsenal's other wing back Alex Oxlade-Chamberlain.
In the 11th minute of extra time, just after Aguero was taken off, Alexis Sanchez scored the winner.
The result gave manager Arsene Wenger, an FA Cup winner six times, a welcome boost in a season that many supporters would like to be his last at the club.
Photo: PHOTOSPORT Arsenal's Alexis Sanchez
Source:
Radio New Zealand