Swansea took control of the tie from the first whistle and raced into a 3-0 inside the opening 25 minutes thanks to two goals from Wayne Routledge and a strike from Michu.
Michael Laudrup's side continued to dominate proceedings in the second half with an own goal from Peterson Pecanha and substitute Alejandro Pozuelo's classy finish completing the rout.
Petrolul Ploiesti had the final word on the night and scored the goal of the game late on with Gheorge Grozav finding the net with an acrobatic overhead kick.
The tie was effectively decided inside 25 minutes as the Romanian cup holders floundered.
There was a hint of luck about Routledge's opener. A Jonjo Shelvey free-kick ran all the way through to the winger but Pecanha looked to have his shot covered until Ovidiu Hoban stuck out a leg to guide it in.
Michu quickly tapped in a second when Pecanha could not hold a Pablo Hernandez shot, before the latter delightfully teed up Routledge for a simple far-post header.
Pecanha then inadvertently provided the fourth as he diverted Wilfried Bony's shot over the line after the Ivory Coast striker's effort squirmed under him, before Pozuelo raced on to a Michu pass to finish in style.
But Petrolul had the last word as Grazov, whose goal had seen off Vitesse Arnhem in the last round, produced a stunning overhead kick to beat Michel Vorm.
The win was the perfect response from Swansea to their disappointing 4-1 defeat to Manchester United last weekend.