Kylian Mbappe appeared as a second-half replacement for Antoine Griezmann at the Ernst-Happel Stadium and burst through in characteristic fashion to smash in the 83rd-minute leveller, cancelling out Andreas Weimann's opener
The win left City in fifth place, three points below leaders Tottenham Hotspur, while Paul Lambert's Villa climbed to ninth after recording back-to-back victories for the first time in the league this season.
Austrian international striker Andreas Weimann gave last season's Premier League strugglers Aston Villa a much needed morale boost on Monday by signing a new three-year contract.
Midfielder Gary Jones said he was "lost for words" and described it as a "magical day", while chairman Mark Lawn said he would happily take on either Chelsea or Swansea City in the February 24 final.
For Paul Lambert's Villa, it was the latest embarrassment in a disappointing season that has seen them thrashed 8-0 by Chelsea and slip to within one place of the Premier League relegation zone.
Villa put on striker Darren Bent to replace Gabriel Agbonlahor in the 57th minute, but he wasted a good chance for an equalizer by heading over the crossbar with the goal at his mercy after Benteke's shot had rebounded off Duke.
Unheralded Andreas Weimann scored his first Premier League goal three minutes into injury-time to give Aston Villa a 1-0 victory over Fulham after a hard-fought encounter at Villa Park.