Any suggestion that Pep Guardiola is not interested in a Manchester City tilt at an unprecedented quadruple was belied by his reaction as Sergio Agüero equalised in the 56th minute.

The goal came from Ilkay Gündogan’s quickly taken free-kick, which displeased Sean Dyche and his assistant, Ian Woan, causing the Manchester City manager to go into hyperactive mode, making what appeared to be yapping actions with his hands at them. If there was any sense the Catalan might not have been too displeased at being knocked out and so lose one of the four competitions City can still win, the passion at levelling this FA Cup third-round tie was the clearest riposte.

During the dispute, Guardiola was told by Andy Haines, the fourth official, to calm down. Asked about the incident and what he told Dyche and Woan, the Catalan said: “I am sorry, I apologise. I didn’t want to lose control. I said: ‘Please leave the referees.’ I have a lot of respect for the way they play – of course, the way they play is completely different than the way we want to play, but they do it perfectly. That’s why football is top. All the managers around the world, especially here, have to do their job. Of course, they [managers] have to keep control. If they were offended, I am so sorry.”

Dyche denied there had been a falling-out with Guardiola. “Not really,” he said. “You both fight, you both want your teams to win. You’re allowed to show your passion, right?”

Guardiola’s declaration he would field a strong XI was borne out by his selection. While there were four changes – Claudio Bravo, Danilo, Oleksandr Zinchenko and Gündogan – David Silva, Leroy Sané, Agüero, Fernandinho and Raheem Sterling were among those in the side. Dyche also brought in a quartet, in Matt Lowton, Kevin Long, Ashley Westwood and Sam Vokes.

City attacked from the opening whistle and won a corner that Sané played short to Zinchenko. Eventually, Sterling let fly, the shot was blocked, and Burnley cleared. For a while, the home side camped inside Burnley’s half and rolled the ball around. When Ashley Westwood fouled Agüero on the edge of the area, City had a free-kick in optimum position. Gündogan smashed this at the wall, though, and Zinchenko skied the rebound.

Some suspect goalkeeping from Bravo followed. Zinchenko passed to the Chilean and, in trying to find Nicolás Otamendi, he hit the ball out for a corner. Johann Berg Gudmundsson fired this in, Bravo was nowhere, and Ben Mee headed back across goal, exposing the one faultline in Guardiola’s team: the rearguard.

When Burnley did this again, they took the lead. Their keeper, Nick Pope, launched a high ball, John Stones miskicked badly, and Barnes roved forward, then smacked a finish past Bravo.

City’s first act of the second half was a raking Danilo ball that hit Sané on a right-left diagonal.

Zinchenko darted into the area, but the wide man overcooked the pass. Later, Sané ballooned a cross straight out, in a further indication that, so far, this was an off-day.

Silva rarely has these and the clever pass that went through Stones’s legs to Gündogan in the area illustrated his usual excellence. Gündogan, though, failed to shoot and Burnley cleared.

The equaliser came from the quick free-kick that led to Dyche’s and Woan’s spat with Guardiola. As this started to calm down, Agüero slotted a 19th goal of the campaign – and 16 in 16 Cup outings – after a delightful Gündogan back-heel that removed Burnley’s rearguard.

Of the Argentinian, Guardiola said: “His quality in the box is amazing and we need his goals. He’s the kind of player who [as today] doesn’t do much in the first half, but, in seconds, he scores two goals. If we want to win something he’s essential for us.”

From here, City coasted, Sané finally finding high gear to slide in for a ninth goal of the season. He then became provider, picking out the substitute Bernardo Silva, who made it 4-1, eight minutes from time.