Anthony Peacock gave Boro the lead in the last quarter of the game and Lee Croft equalised within minutes, but neither side could break the deadlock and the game went into extra-time.
Andrew Davies rose to head home a corner half way through the first-half of extra-time and Boro hung on as City threw caution to the wind, goalkeeper Ross Turnbull keeping Boro ahaed with a fine save from Croft.
Mark Proctor's team will now meet either Manchester United or Charlton in the final. Their semi-final in evenly poised at 1-1 with the second leg to be played at Old Trafford next Wednesday.