A little after 8.15pm Mestalla broke into applause but it was Swansea they clapped, not Valencia. Jonathan de Guzmán had just scored a glorious free-kick to put the visitors 3-0 up in this first Europa League group game and completed a night that some dared to rank among the best in their history. "It was a fantastic game and I am very pleased," Michael Laudrup said, "but I cannot say that: I have only been here for 15 months: you would have to ask the people who have followed Swansea their whole lives."

Those who travelled to Spain's east coast certainly enjoyed it and the home fans could not help but be impressed. If there was opprobrium for Valencia there was genuine admiration for Swansea. A chant went round declaring the home players were unworthy of the shirt but Swansea had honoured theirs. This is their first European campaign since 1991 and, having come through two qualifying rounds, this was a big victory. In a group that also contains St Gallen and Kuban Krasnodar, the two favourites for first place competed at Mestalla but it was no competition.

"No one expected us to get three points here but it is only one game so I do not want to talk about being favourites in anything," Laudrup warned. He is right; it was just one game. But what a game, and Swansea are already well placed to progress. It was the perfect start. "I am pleased for the fans: they have had a great night, one to remember. But I am happy for the players too, they have seen that they have no problems going away and playing against big teams in Europe," the Swansea manager said.

Swansea, who started with six Spaniards to Valencia's four, finished the match with 19 shots to Valencia's six and that had been made more respectable by a late but largely impotent flurry. The shots had been Swansea's and the possession of the ball had too. Laudrup said his team had "dominated from the first minute to the last". Their opponents had contributed to their own demise, almost from the start.

The moment that helped to define this game came after nine minutes and it began, as much of the danger Valencia faced would begin, with a gift. Ever Banega, playing deeper than normal, gave the ball away. Wilfried Bony was the swiftest to react, dashing through the middle, and, although he was still 15 yards outside the area when Adil Rami brought him down, there were few complaints when Serge Gumienny pulled out the 100th direct red card in the Europa League: the foul looked deliberate and Bony had a clear run on goal.

Valencia's coach, Miroslav Djukic, soon removed his left-winger Fede Cartabia and shifted to something approaching a 4-4-1. If he sought security, he did not get it. Ricardo Costa came on and into the middle of the defence and the first time he touched the ball was when it glanced off his outstretched leg and into the net after he failed to block Bony's shot. The goal from Bony was provided by Michu, his clever cut-back from the left finding the striker who hit it first time.

Michu celebrates scoring Swansea City's second goal against Valencia. Photograph: Adam Holt/Action Images

Before the match Valencia's full-back Andrés Guardado had, like José Mourinho, been talking eggs. The difference is that in Spain the word for "eggs" is also slang for testicles. Guardado demanded his side have balls but the ball was another matter entirely. Valencia could not get possession and when they did, they tended to lose it in compromising positions. Slowly they were falling apart. Swansea were comfortable now. They took complete control, racking up 66% of possession in the first half, never looking remotely perturbed.

Antonio Barragán's mistake set Nathan Dyer through but he shot wide; Javi Fuego's error saw Bony bulldozing through again, sending Jérémy Mathieu and Costa crashing into each other. Djukic had talked about Valencia's "vulnerability" and it was painfully evident here. Before half-time Michu headed over. The respite would be temporary.

Valencia had shot once, Swansea 11 times. Within five minutes of the restart Bony had two more, provided by Dyer and Michu, then Michu sliced a shot over the bar on the bounce. Momentarily Valencia appeared to awaken when Guardado set off on a run but instead it was a prelude to Swansea's second and effectively the end of this game. "When we scored the second we had already had four clear chances to score," Laudrup said.

Possession was won back and the counter-attack began. They progressed steadily but almost in slow motion, the move based more on precision and patience than pace. De Guzmán carried forward, the ball was spread wide to the right and back inside again, where Alejandro Pozuelo waited for Michu to make the run beyond him and beyond Mathieu. The pass was rolled perfectly into his path and Michu finished low at the near post.

Soon afterwards De Guzmán swung a wonderful free-kick into the top corner from 35 yards and Mestalla broke into an ovation. When Michu was withdrawn a little later there was more appreciative applause. He and Swansea had earned it. This was their night.