Juan Mata scored the winner as Manchester United bounced back from their humiliating defeat at Chelsea with a 1-0 derby victory over Manchester City in the EFL Cup fourth round, extending their local rivals' winless run to six matches.

This was a tie between two managers in need of a reassuring win – Jose Mourinho after the embarrassment of a 4-0 defeat on his return to Stamford Bridge on Sunday, and Pep Guardiola amidst what is now the longest barren spell of his coaching career.

A much-changed City side started strongly but were unable to make the breakthrough, allowing United to find a way back into the match.

After finishing the first half strongly at Old Trafford, the hosts made a flying start to the second, hitting the post through Paul Pogba before Mata netted the winner.

The victory sees United into the quarter-finals and lands a satisfying blow for Mourinho and the red half of Manchester on Guardiola and outgoing EFL Cup champions City.

Guardiola made nine changes to the side that drew at home to Southampton on Sunday and it was Kelechi Iheanacho – in for Sergio Aguero – who got the first sight of goal in the second minute.

The Nigerian striker headed over from close range after getting on the end of an inviting Jesus Navas cross, and City continued to probe a United defence that lacked the injured Eric Bailly, seeing claims for a penalty waved away when the legs of Aleix Garcia and Michael Carrick tangled in the box.

Claudio Bravo's well-publicised lapses in recent matches may have given Willy Caballero hope that he could unseat the City number one with a strong performance at Old Trafford, but neither he nor David De Gea in the opposite net were given too much to do in a first half that ended without either side having registered a shot on target.

City looked the more likely to break the deadlock for much of the opening period, but it was United who finished the stronger, although Zlatan Ibrahimovic sent a promising free-kick straight into the wall before having another effort from the edge of the box deflected over.

Vincent Kompany, who only returned from long-term injury at the weekend, did not emerge for the second half – replaced by Aleksandar Kolarov – and it took United just a few minutes to find a gap in the adjusted City back line.

Played through on goal by Ibrahimovic, Pogba's strike was brilliantly tipped onto the left-hand post by an alert Caballero.

United had their tails up and got the goal their early second-half pressure merited in the 54th minute, when Ibrahimovic's squared pass was side-footed home by Mata – the goal allowed to stand despite a coming together between Ander Herrera and Fernando in the box.

City continued to struggle to carve out clear-cut opportunities and United came close to a second when Ibrahimovic just failed to connect with a well-placed Marcus Rashford cross.

The late introduction of Sergio Aguero and Raheem Sterling failed to reap any reward for Guardiola, who cut a frustrated figure on the touchline as hope of an equaliser faded.