Burnley led 1-0 but had conceded a free kick, 40 yards from their goal.
City boss Guardiola defended his actions, claiming he was merely standing up for the match officials.
The result means Burnley are knocked out of the competition, while Premier League leaders Manchester City will have to wait and see who they will be drawn with in the FA Cup fourth round after 7pm on Monday.
Aguero quickly added a second and Dyche added: "We got a little punch drunk from that moment and that's definitely a learning curve for the future".
Yet despite that, Guardiola named a strong side.
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"It's an honour for me to have Aguero under my command".
The game also saw David Silva return from Spain where he was tending to his prematurely-born son. The efforts of the Spanish midfielder, who has been at the club for eight years, were appreciated by the fans who gave a warm tribute of applause in the 21st minute.
Dyche said: "I heavily question a moment that changed the course of the game".
"I am happy to be part of this team and train hard every day, waiting for my chance".
With the Premier League as good as won, every passing week has more observers, and many more City supporters, jumping on the quadruple bandwagon.
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While it was an impressive piece of play, the best was to come just two minutes later.
Even if you don't support them, you have to sit back and appreciate Aguero and Gundogan's contributions.
For about 55 minutes City looked like a team thinking they could actually lose a game. Fatally, the Burnley keeper hestitated, allowing Sane to waltz past him and set up Bernardo Silva for a simple finish. "I needed time and patience, but it has already happened and now everything is okay". And the Catalan believes they have made a good start.
"This is only my second year in Europe - before that I was in Russian Federation and Ukraine and they play very different football".
"The timing was just flawless", said Gundogan.
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