Chelsea led for most of their fifth-round match against fellow Premier League side Bournemouth, but a late goal from Dan Gosling appeared to be sending the match to extra time, only for an Alvaro Morata stoppage-time winner to ensure the Blues's passage into the semi.
Arsenal found things a little more comfortable with a 1-0 win in another London derby, against West Ham United, on Tuesday.
Bristol City, currently sitting third in the Championship, are the surprise team in the draw. The only non-Premier League side in the quarter-finals, they snuck through with a stoppage time winner by Korey Smith to stun the Red Devils and knock the reigning Carabao Cup champions out of the competition.
Man City also saw drama in their quarter-final match, having it come down to penalties against Leicester City as they kept their quadruple hopes alive for the season.
The teams will meet in a two-legged play-off, with the first leg scheduled for January 9 and 10 and the return fixtures January 23 and 24.
The winners will contest the final at Wembley Stadium on February 25.