Michael Laudrup's men fancied their chances heading into the Group A encounter as Valencia have made a poor start to the season under new manager Miroslav Djukic, losing three of four games in La Liga.
And they got a massive boost on 10 minutes when defender Adil Rami saw red for hauling down Swansea striker Wilfried Bony, who had pounced on a defensive slip and was through on goal.
It did not take the visitors long to make the dismissal count as they took the lead on 14 minutes, when Michu broke down the left flank and pulled the ball back to strike partner Bony, who thumped a deflected left-footed strike past Che goalkeeper Vicente Guaita.
Bony sliced another shot wide on 39 minutes as the Swans controlled proceedings up to half-time, leading to a protest from a large number of home fans who deliberately missed the early stages of the second half.
During that time Bony steered a header from Nathan Dyer's cross over the crossbar and Michu's left-footed strike from 10 yards cleared the bar too.
But the fans were back in their seats to see Alejandro Pozuelo thread a fine throughball to Michu, who clinically slammed a right-footed shot past Guaita to make it 2-0 on 58 minutes.
It was 3-0 four minutes later as Jonathan de Guzman curled a stunning 30-yard free-kick into the top right-hand corner, and with Chico a rock at the heart of Swansea's defence, the visitors eased through the closing stages to keep a clean sheet on an historic night for the Welsh outfit.
Valencia are in the midst of their worst run in 14 years - but even the most optimistic Swansea fan could not have anticipated such a great start.
The Welsh side, fielding more Spanish players than their hosts, took control after Valencia were reduced to 10 men with less than 10 minutes on the clock.
Rami dragged back Bony as the striker got away from him 30 yards from goal. The French international had team-mates near but Belgian referee Serge Gumienny did not hesitate as he produced a red card.
Los Che boss Djukic was forced to sacrifice Fede Cartabia to get defender Ricardo Costa on, but seconds after he made the change his side conceded.
Pozuelo fed Michu on the Swansea left, and when his pull-back ran through to Bony the striker dispatched a left-footed strike high into the net with the aid of a deflection off Ever Banega.
Swansea were in complete control, dominating possession as their numerically disadvantaged hosts toiled.
Bony's power was providing them with a particular headache, and the Ivorian almost got his second of the night five minutes before the break, getting the better of Costa before fizzing a shot narrowly wide.
The half-time whistle was greeted with boos from the home faithful and a few white hankies for the under-fire Djukic.
Some had seen enough, with the lower tier behind one of the goals - which had been full - slowly emptying as the teams returned to the field as part of a planned protest, before the fans reclaimed their seats around 10 minutes into the second period.
In that time, Swansea missed two gilt-edged chances. Bony headed over from close range after an excellent Dyer cross, while Michu volleyed over when unmarked.
When the supporters did return, they were just in time for Swansea's second goal. De Guzman laid the ball off to the lively Pozuelo and his lovely pass afforded Michu the room to fire home with his right foot.
It was soon 3-0. Pozuelo was fouled by Banega 30 yards out, and De Guzman stepped up to produce an unstoppable curling, dipping free-kick that beat Guaita all ends up, handing Swansea just a second away win in European competition.
|Valencia||Team Statistics||Swansea City|
|0||1st Half Goals||1|
|2||Shots on Target||6|
|4||Shots off Target||8|
|30.3||1st Half Poss.||69.7|
|28.7||2nd Half Poss.||71.3|