The Reds' defence of the trophy came to an end after conceding a stoppage-time to goal to lose 2-1 at Ashton Gate.
"It was a brilliant day for football - congratulations to them".
Jose Mourinho's side were holders after winning the League Cup last season.More news: Nashville expected to be named MLS expansion franchise
Mourinho said: "They were a bit lucky but they fought a lot to be lucky".
"That's why they are one of if not the biggest club in the world". They deserved a place in the semi-finals against Manchester City for their defensive organisation, for their hunger and for taking their chances.
"But I don't want to blame the players at all because I think they were professional - it wasn't a case of "we don't care about the game" or "we don't have responsibility" or "we don't have pride".More news: Facebook challenges music streaming giants with Universal Music deal
Lee Johnson has praised his Bristol City players for each doing "a man and a half's job" while knocking Man Utd out of the Carabao Cup.
"Do the recent results against them give us extra motivation? Yes". Everyone was waiting for our goal so they were lucky.
Police apprehend fans as they invade the pitch after the match against Manchester United. Chelsea hosts Arsenal on January 9 at Stamford Bridge and the second leg will be at Emirates Stadium on January 23, while Manchester City will play the first game at home on January 10 and the second leg will be played in Bristol on January 24.More news: Twitter begins enforcing rules on 'hateful, abusive' content
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