The home side were forced to play 80 minutes with 10 men after defender Adil Rami saw red for hauling down Wilfried Bony.
The striker went on to give Swansea the lead in the 14th minute with his fifth goal in eight games, firing home with the help of a deflection.
Michu and a wonderful Jonathan de Guzman free-kick completed the scoring in the second half as the Swans cruised to victory at a stunned Mestalla.
There have been more important victories in Swansea's 101 years, but there will have been few as eye-catching as this, adding another remarkable chapter to their remarkable rise from League Two to the Premier League, Capital One Cup glory and this European campaign.
Valencia, two-time Champions League finalists, had only lost to English opponents on three previous occasions here, but they were in trouble from the moment Rami saw red.
Bony threatened to race clear of the Frenchman, and the referee quickly had the card out of his pocket as Bony went to ground.
The Ivory Coast international got the opener shortly after as Alejando Pozuelo and Michu combined for him to rifle home.
Pozuelo, an exciting presence throughout, was involved again when his excellent pass teed up Michu for the second in the 58th minute.
And De Guzman rounded matters off with a spectacular strike four minutes later to leave the Welsh club in dreamland.
Source: PA
Source: PA