Bristol City were "lucky" to beat Manchester United but the result was a "beautiful day for football", said Red Devils manager Jose Mourinho.
Korey Smith scored an injury-time winner as the Championship side beat the Carabao Cup holders 2-1.
The Robins will face unbeaten Premier League leaders Manchester City in the two-legged semi-finals.
"A team from the lower division won. A big day for them," Mourinho told Sky Sports.
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The Robins reached the last four of the competition for the first time since 1988-89 with a well-deserved victory at Ashton Gate, going ahead in the second half through Joe Bryan's thunderbolt into the top corner.
United had already hit the crossbar and post - and Zlatan Ibrahimovic equalised with a deflected free-kick.
But Smith's cool strike sparked jubilant scenes as United became City's fourth top-flight scalp in this year's competition, after Watford, Stoke and Crystal Palace.
Mourinho said: "They were a bit lucky but they fought a lot to be lucky. Everyone was waiting for our goal so they were lucky. We hit the post twice.
"But they played brilliantly, they fought like it was the game of their lives which probably it was. A beautiful day for football."
United players blow their chance
Mourinho, who has won the competition four times, made 10 changes to his starting line-up after beating West Brom on Sunday, but still included Paul Pogba, Anthony Martial and Marcus Rashford.
The former Chelsea boss suggested some of those who were named in the initial 11 had blown their chance to impress.
"The reality is that the players that were on the pitch were the ones that were not in the last match," said Mourinho.
"They are the ones that are not going to be in the next match. I think they lost a good opportunity also to play semi-finals and also to have two more matches to play.
"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'.
"I think they showed they had that after the 1-0 [the first goal] and they had that until they scored the second goal in the last few seconds and we had no chance to react."
Analysis - No wine for Mourinho
BBC Sport's Tom Garry at Ashton Gate:
After the final whistle, Mourinho rushed through the media zones so quickly that he delivered - bizarrely - his media conference when only one member of the written press was in attendance.
Many colleagues commented that they had never seen a manager leave so quickly after a televised game. When he did speak, the Portuguese referred to Bristol City as "lucky" numerous times.
However, before the game, he was good enough to spend more than 15 minutes giving managerial advice to 36-year-old Robins head coach Johnson in private, who clearly hugely appreciated that gesture.
Johnson, who is enjoying the best season of his young career so far, had hoped the pair would share a bottle of wine after the game, which had cost the Robins boss £450 to order especially from Portugal.
But the taste of success will be far sweeter for the Championship club, who have now beaten four Premier League sides in the competition already so far this season.