An evening that threatened to be problematic for Swansea City, as well as freezing cold, ended up delivering a routine victory for the Premier League club and a place in the FA Cup quarter-finals for the first time since 1964. Tottenham Hotspur or Rochdale will visit the Liberty Stadium a fortnight on Saturday after second-half goals from Jordan Ayew and Nathan Dyer enabled Swansea to rack up a sixth successive home win at the expense of Carlos Carvalhal’s former club.

Sheffield Wednesday created the better chances in a thoroughly underwhelming first half but Swansea’s class eventually told on a night when the home supporters who braved the elements were treated to free cups of tea and coffee. The football took a while to warm their hearts but the arrival of Ayew helped in that respect as the Ghanaian came off the bench at the start of the second half to score his ninth goal of the season.

Wednesday, 17th in the Championship, never really looked like recovering from that setback and any hopes they had of causing an upset were extinguished 10 minutes from time when Tammy Abraham released Dyer on the right and the winger slid the ball through Cameron Dawson’s legs for Swansea’s second goal.

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Carvalhal was quick to stress afterwards that Premier League survival remains the Welsh club’s priority but the 52-year-old was his usual enigmatic self when asked whether he would allow his players to dream now that Swansea are 90 minutes from Wembley. “In my jacket it says: ‘Carlos has a dream,’” said the Portuguese, pointing to the words embroidered on the inside of his blazer. “All of us have dreams. We must dream of something.”

Swansea have already played enough matches this season to win the FA Cup – this was their sixth game in the competition and their third home replay – but it has now reached the stage where Carvalhal will surely want to put out his strongest side rather than rotate. He made five changes against Wednesday and the truth is that Swansea struggled until the introduction of Ayew, their leading goalscorer, gave them a legitimate threat up front.

With their passing slow and pedestrian in the first half, there was no tempo to Swansea’s play and an alarming lack of creativity further forward, where Abraham made little impression as the focal point of the attack and Wayne Routledge, the player who made way for Ayew at the interval, was a substitution waiting to happen.

Wednesday also started poorly but the visitors grew into the game and had a couple of opportunities around the half-hour mark. Jacob Butterfield unleashed a powerful 22-yard drive that was heading for the far corner until Kristoffer Nordfeldt superbly turned the ball away with his right hand and moments later the Swansea goalkeeper was scrambling across his line to push Lucas João’s cross-turned-shot beyond the far upright.

Swansea did little more than huff and puff at the other end. Sam Clucas tried a spectacular overhead kick that was much further away from the goal than a sparse crowd seemed to realise and Ki Sung-yueng drilled a low left-footed shot that brushed the sidenetting. That was as good as it got for Swansea during the opening period.

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Carvalhal was clearly unimpressed and made a double change at the interval that brought instant reward after Tom Carroll thumped a left-footed shot from 25 yards that cannoned off the feet of both posts. Ayew was quick to pounce on the loose ball and steered home via a slight deflection off Glenn Loovens.

Swansea were now in control of the game and that superiority was reflected with their second goal as Wednesday, who never played with enough belief or conviction, were carved open. Ayew linked up with Abraham and the forward picked the right moment to feed Dyer, who dispatched a low shot past Dawson.

“I am very happy and very proud of my players,” Carvalhal added. “I was born in 1965. The club didn’t do better than this in the Cup since 1964, so it’s been a long, long time. I am happy we have achieved this. Now we will play either Rochdale or Tottenham at home and we could go to Wembley, so we are in a very good position.”