City took on the South Yorkshire outfit at Wembley looking to end a run of three consecutive defeats at the semi-final stage.
Sheffield United, closing in on a Premier League return, proved dogged opposition and held out for 40 minutes before conceding a penalty.
Riyad Mahrez scored from the spot - the first of an historic hat-trick - and his second was a sublime solo effort.
City never looked back from that point, booking a final place at Wembley against either Brighton or Manchester United.
Watch the best of the game by playing the video above...