The home side enjoyed the better of the first half then just as the visitors were beginning to find a foothold of their own late in the match, Graham struck with four minutes remaining as Liverpool’s poor away form let them down again.
Swansea started by playing the attractive passing game that has typified their season and won them so many new admirers, a weary-looking Liverpool side struggling to get a touch in the opening 10 minutes.
The first chance of the match duly fell to Swansea, Joe Allen nicking the ball off Jonjo Shelvey before delivering a cross just wide of the onrushing Graham who was inches away from a tap-in that brought back memories of Gascoigne against Germany.
A poor back pass from Andy Carroll saw Alexander Doni inexplicably pick the ball up under minimal pressure from Graham to give Swansea a free-kick five yards inside the Liverpool penalty area - but Scott Sinclair saw his shot blocked by Jamie Carragher.
Doni made amends for his lapse in concentration shortly afterwards, saving Gylfi Sigurdsson’s free-kick wonderfully low down to his right. And the Liverpool keeper again had to be at his sharpest to keep out Nathan Dyer just before the break, saving well with his legs after Sigurdsson played in the winger.
Liverpool’s chances were few and far between, limited to sporadic pot shots from Carroll and Shelvey on the edge of the box and a fractious end to the half saw Carroll and Ashley Williams both booked after a scuffle that ended with the Liverpool striker’s hands around the neck of the right back.
Swansea continued to create chances after the break, Doni again saving well from Dyer. But it was Liverpool who went closest to breaking the deadlock, Michel Vorm saving spectacularly from Carroll’s acrobatic bicycle kick. And Liverpool should have gone ahead when Carroll found himself with a great chance, only to scuff his effort into Vorm’s grateful hands.
The game looked headed for a stalemate but Swansea continued to press and deserved their goal on the balance of play. The win lifts them to 11th in the table while Liverpool finish eighth after a season many of their fans would rather forget.