The benches were still arguing when Matt Lowton lost possession outside his own area and Aguero could pick out Gundogan and run on to the back-heeled return to score the second.
The visitors barely had time to regroup before Gundogan’s clever flick picked out Aguero, while Sane prodded in the third and Bernardo Silva added the fourth to settle matters.
This will be the first meeting of any kind between these sides since Man City won a second round League Cup tie 2-1 at Ashton Gate in August 2007 – Emile Mpenza and Rolando Bianchi scoring for Sven-Goran Eriksson’s City.
“I’m so happy”, said Guardiola.
In contrast, Burnley looked a threat on the rare occasions they broke forward though they should give Stones the assist for taking the lead after 25 minutes.
David Silva, who has been travelling to and from Valencia following the premature birth of his son Mateo, was also back in Manchester for the game.
He set up Leroy Sane for the third goal and received a minutes’ applause from City fans in the 21st minute as they showed their support for the midfielder and his family. It was like he’s never been away.
That flashpoint was a rare moment of concern for Guardiola and City on an afternoon in which they stormed back after trailing to Ashley Barnes’ first half goal.
A long ball upfield was flicked on by Sam Vokes before being completely missed by John Stones, allowing Barnes to capitalize on the error by racing through on goal and slamming his finish past the keeper despite the angle being against him.
That defeat ensured City boss Pep Guardiola would end his first season in English football without a trophy – the first campaign he had failed to secure silverware in his managerial career. Does he not play any more?
In the 56th minute, Gundogan’s quickly-taken free-kick was perfectly delivered to capitalise on Aguero’s run and the City striker drove home with his right foot.
Manchester City fielded a strong team for this cup tie despite making four changes from their midweek league game against Watford which included Claudio Bravo starting between the sticks. City were by no means at their worst, Fernandinho and Sane missing good first-half opportunities, but Burnley’s physical approach has a way of disrupting the top sides.
As expected, City’s dominance came through in the second half although it still took some quick thinking from Gundogen and Aguero to break the Clarets’ resistance. It wasn’t for Sean Dyche.