Reds boss Kenny Dalglish has already had to deal with reports that he faces a fight to save his job after an underwhelming league campaign and this lacklustre effort was hardly a ringing endorsement of his reign.
The club's American owners will conduct a wide-ranging review of Liverpool's season, which started with hopes of challenging for a top-four finish but ended with the team languishing in eighth spot.
Liverpool couldn't even take the slender consolation of finishing above Everton, who ended up four points above their Merseyside rivals.
Winning the League Cup against Championship club Cardiff at least ended the club's wait for silverware, but that was followed by a tepid performance in the FA Cup final defeat against Chelsea.
Swansea manager Brendan Rodgers had encouraged the home fans to turn up in Elvis Presley costumes to celebrate their survival after one critic suggested there was more chance of a sighting of the deceased singer than Swansea staying up.
The party mood inspired the hosts and Graham came close to giving them an eighth-minute lead but could not convert Joe Allen's cross-shot after the midfielder had robbed Jonjo Shelvey.
Maxi Rodriguez was off target with a half-hearted effort for the visitors but the majority of the action was taking place in the Liverpool half.
Gylfi Sigurdsson was left unmarked at a corner and almost swept home from Allen's pinpoint delivery but his effort was bravely blocked by the onrushing Liverpool defence.
Sigurdsson was brilliantly denied by Doni in the 32nd minute, as the Iceland international intelligently struck a free-kick across the Liverpool wall, but the Brazilian showed lightning reactions to tip the ball round the post.
Andy Carroll wasted Liverpool's best chance when he fired narrowly over Vorm's bar five minutes before the break.
Carroll's increasing frustration was clear to see and he was involved in a tussle with Ashley Williams which earned the pair bookings.
Dalglish sent his players out early for the second half. Initially his team-talk seemed to have the desired response as Carroll's far-post header was diverted to safety off Steven Caulker's back before the England forward produced a brilliant overhead kick that Vorm matched with an agile stop.
Carroll's central role in proceedings continued as his defensive mistake almost allowed Rangel to poke home.
But the hosts bagged the winner in the 86th minute as Angel Rangel and Scott Sinclair linked on the right and Graham converted his 14th goal of the season.
Source: AFP
Source: AFP