The Championship club won 2-1 at Ashton Gate to set up a semi-final against Manchester City.Korey Smith’s 93rd-minute strike settled the Robins’ first meeting with United since 1980.
“It was an unbelievable moment. We want more nights like that,” head coach Lee Johnson told BBC Sport.
Bristol City players did not know whether to “laugh, cry or cuddle each other” after beating Manchester United in the Carabao Cup quarter-final.
“With the noise in the stadium and the emotion, the boys didn’t know whether to laugh, cry or cuddle each other.”Johnson, 36, ran down the touchline to celebrate Smith’s late winner, picking up and hugging a ball boy amid scenes of joy at a sold-out Ashton Gate.
“I didn’t know what to do,” Johnson added. “I wanted to run and celebrate with the lads but I couldn’t do that. He (the ball boy) was the only one there.
“It was great to see him because I needed to celebrate with somebody and it just so happened it was the ball boy.”
Victory was Johnson’s side’s fifth in a row in all competitions and their fourth against Premier League opposition in this season’s EFL Cup.
Before the last-gasp drama, goalkeeper Luke Steele’s saves had kept the scoreline at 1-1 after Zlatan Ibrahimovic’s equaliser for United had cancelled out Joe Bryan’s stunning opener.
Bristol-born left-back Joe Bryan was hailed as a “Champions League player” by Johnson after his display in Wednesday’s win.
A product of the Robins’ youth academy, Bryan shone in the club’s highest-profile match since the 2008 Championship play-off final.
The 24-year-old – who can also play at left midfield or wing-back – opened the scoring with a powerful drive in to the top corner after a slick team move that he started with a clever touch.
“Joe Bryan is a Champions League player, without question,” Johnson told BBC Radio Bristol.
The former Barnsley boss continued: “I can honestly say Joe’s done that 20 times in the past four or five days in training.
“Showing the technique to keep a straight core, tighten that straight back and just keep his head still and fire through it, he’s a top player.”
Former Bristol City and Sunderland midfielder Gary Owers, co-commentating for BBC Radio Bristol, added: “That was a world-class performance from a local boy who has come through the academy.”
Johnson’s side will face unbeaten top-flight leaders Manchester City in the last four in the week beginning 8 January, before the second leg at Ashton Gate two weeks later.