Wolves doubters were out in full force in the buildup, with many backing Bristol City to keep their recent history of FA Cup upsets going. However, the sceptics were silenced as Wolves displayed top-flight class in a much-improved display.

Related: Chelsea v Morecambe, Leicester v QPR, and more: FA Cup third round – clockwatch

Vítor Pereira was predicted to make plenty of changes but he settled on three from the club’s 3-0 Premier League loss to Nottingham Forest with the goalkeeper Sam Johnstone, André and the new signing Emmanuel Agbadou coming into the starting lineup. The arrival from Reims had a good debut after joining on a four-and-a-half-year deal on Thursday. He was calm and assured on the ball, won free-kicks and confidently marshalled the backline.

On his new signing, who is a natural centre-back but played on the left, Pereira said: “He was consistent. To play on the left side without a left foot is not easy because you press the best foot of them. He can play with left and he played well in my opinion.

“He realised today, the game never stops here. There are fast free-kicks, fast throw-ins, when the goalkeeper starts to play, the ball never stops. The players don’t have the time to recover and to breathe a bit.”

Despite early dominance from Wolves it was City, who made three changes themselves, who had the first chance. Scott Twine won a free-kick and took it himself but the effort was straight at Johnstone.

Rayan Aït-Nouri had Wolves’ first chance but Max O’Leary was wise to it and made a good save. Wolves kept the pressure on as they won the first corner of the game. It was initially repelled but João Gomes sent the ball back in and this time Aït-Nouri didn’t miss with a bullet header.

There were protests from O’Leary as he was convinced the Algerian handled the ball but the officials did not intervene. The video assistant referee system is not in operation this season until the fifth round of the Cup.

The away end, who had been pretty vocal before the goal, cranked up their noise as “Oh … Vítor Pereira” rang around Ashton Gate. Wolves supporters only got louder as the club doubled their lead. Slick passes between Jørgen Strand Larsen and Gonçalo Guedes opened space for Rodrigo Gomes to finish calmly.

Bristol City grew into the game, particularly by creating more chances for Fally Mayulu, and their patience paid off. The hosts won a free-kick in a great position and Twine stepped up once again, scoring a sensational goal which sent the stands wild. Not only did it liven up the home support but it kept their hopes alive as they headed into half-time.

• Download the Guardian app from the iOS App Store on iPhone or the Google Play store on Android by searching for 'The Guardian'.

• If you already have the Guardian app, make sure you’re on the most recent version.

• In the Guardian app, tap the Menu button at the bottom right, then go to Settings (the gear icon), then Notifications.

• Turn on sport notifications.

There was bad news for the Robins in the second half, however, as Marcus McGuane came off after lengthy treatment. The Bristol City manager, Liam Manning, said: “We will have to assess, he took a whack to the head so we will wait and see.”

Chances came at both ends, the best coming from a Bristol City free-kick with Ross McCrorie hitting the crossbar. The ball bounced into the goal area but Wolves frantically cleared to hold on for victory.

Pereira said the priority was the Premier League. They are 17th, despite the Portuguese’s reign starting with two wins and a draw, and said naming a strong side here was integral to that aim, with only four days until they visit Newcastle. “We don’t have time for training and I want to increase some behaviours, it is important to use the games to do it,” he said. “I think we will have time to recover for the next game. I really believe with the players who are injured now, when they recover and we have new players in specific positions, we have a good team and squad to be in the Premier League next year …

“I don’t want to say it was a good training because [it can come across] bad. But we need to play together and to play the way I want to increase our level.”