City boss Guardiola gestured angrily towards Dyche, making a 'yapping' gesture at his counterpart and the fourth official, as a furious row erupted between the two at Eastlands on Saturday following Sergio Aguero's 56th-minute equaliser.
"I am sorry, I apologise. I didn't want to get out of control," said Guardiola.
Aguero's leveller came from a quickly-taken Ilkay Gundogan free-kick which angered Burnley players, who thought that referee Graham Scott allowed the set piece to be taken before they were ready.
Guardiola later claimed his gestures were simply a request for Burnley to leave the officials alone, but it was certainly an ugly scene and one that ended with fourth official Andy Haines ordering the Spaniard to return to his technical area.
"I said 'leave the referees please'. I have a lot of respect for what he (Dyche) does and the way he plays is completely different to how we want to play. But they do it perfectly and that is why football is top. I think all the managers around the world have to control. If they were offended, I am sorry."
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.
Aguero's brace, which came inside 94 seconds, plus further efforts from Leroy Sane and Bernardo Silva carried City through to the fourth round of the cup and left Guardiola singing the praises of City's all-time record goalscorer.
"First of all I have to say it’s an honour for me to have Sergio Aguero under my command," he said. "He is always the kind of player who always wants to play and I can understand sometimes he is sad if he doesn't play. I want him to be happy. The quality he has inside the box is amazing and both of his goals were amazing. He is an essential player for us if we want to win something."
The game also saw David Silva return from Spain where he was tending to his prematurely-born son.
The City midfielder only came back to Manchester on Friday afternoon, less than 24 hours before the cup tie.
Guardiola revealed that Silva was due to return to Valencia immediately after the game and continue to "commute" between Spain and England while his son receives treatment.
The Spanish international released a statement earlier this week to confirm the reasons why he was spending time in his home country.
"I'm happy he made the statement," said Guardiola. "It's not easy to go to Spain, come back and play like that. Now he's going back to Spain and he'll be back when he feels good, we are happy because everything is going well. David is a legend for the club, what he's done."
Dyche laughed off suggestions that there was any ill-feeling with Guardiola over the equaliser but was angered by the officiating that led to the City goal.
"You both fight, you both want to win, you both want your teams to win," he said. "That's how it should be. I believe everybody is allowed to be passionate. The referee said he didn't have to blow the whistle, but Ashley Westwood was five yards away and the ball went through the gap where he should have been. Sometimes in football the rule may be the rule but there's a common sense moment, we've just allowed it to be passed through, I'm bound to be aggrieved by that."
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