The Swans were forced to settle for a point in a remarkable eight-goal thriller at the Liberty as they confirmed their Premier League status for next season.
Butalready-relegated Wolves came back from 4-1 down after 31 minutes to share the spoils in a spirited fightback the visitors.
Andrea Orlandi got the scoring underway as he headed home after just 24 seconds - the fastest goal in the Premier League this season.
Joe Allen doubled the lead just three minutes later before Nathan Dyer headed home from Orlandi's cross.
Steven Fletcher pulled one back before the half hour but the Swans regained the three-goal lead when Danny Graham scored his 13th goal of the season to make it 4-1.
Wolves then raised their game further in the second half and when David Edwards made it 4-3, it set up a tense finish.
The visitors then grabbed the equaliser as Fletcher's cross was tapped in by Jarvis with 20 minutes to play.
For the first time this season, the Swans went with a 3-4-3 formation as skipper Garry Monk returned to the starting XI in a defensive trio along with Ashley Williams and Steven Caulker.
In came Andrea Orlandi as part of two changes to the team that drew at Bolton the previous week, while Angel Rangel (hamstring) missed out and fellow full-back Neil Taylor was among the substitues.
Wolves, already relegated from the top flight, handed Dorus de Vries his second Premier League start as the Dutch keeper returned to the Liberty for the first time since leaving the Swans last summer.
Swans: Michel Vorm, Garry Monk (capt), Ashley Williams, Steven Caulker, Leon Britton, Andrea Orlandi (Neil Taylor, ht), Joe Allen, Gylfi Sigurdsson, Nathan Dyer, Scott Sinclair, Danny Graham (Luke Moore, 82).
Subs: Gerhard Tremmel, Mark Gower, Josh McEachran, Wayne Routledge, Leroy Lita.
Wolves:Dorus de Vries, David Edwards, Richard Stearman, Michael Kightly, Karl Henry (capt), Steven Fletcher (Sylvan Ebanks-Blake, 76), Stephen Ward, Christophe Berra, Matt Jarvis (Stephen Hunt, 72), Kevin Doyle, Kevin Foley.
Subs: Carl Ikeme, Roger Johnson, Nenad Milijas, Eggert Jonsson, Ronald Zubar.
The Swans got off to a dream start as after just 24 seconds they opened the scoring.
Wolves had not even had their first touch of the ball as Andrea Orlandi found Scott Sinclair and the winger returned the favour by crossing to the Spaniard, whose header looped over Dorus de Vries and into the net.
It was a terrible start for the visitors, but it got worse for Terry Connor's side as the second goal arrived with just four minutes on the clock. In a neat passing move which started with skipper Garry Monk, Danny Graham and Gylfi Sigurdsson linked up before the Icelandic midfielder's first-touch pass sent Joe Allen in on goal.
Allen raced forward and his shot beat the wrong-footed de Vries after taking a deflection off Richard Stearman.
"The Jacks are staying up" was the chant coming from the Swans supporters, and they had more to sing about after 15 minutes.
Orlandi nutmegged Kevin Foley and then his deflected cross landed nicely for Nathan Dyer, who headed into the near corner of the net.
Things could have got even worse for Wolves as Graham found his way in on goal after a miss-pass fell to his feet. But de Vries covered his bases as he raced out and stretched to deny the striker's goal-bound effort.
The stunned visitors looked to form some sort of response, and they did well to create an opening as Matt Jarvis' superb low cross found Kevin Doyle, but Ashley Williams perfectly-timed his challenge to clear the danger from eight yards out.
Then, with 29 minutes on the clock, the visitors grabbed a goal back. Doyle's pinpoint cross found Steven Fletcher in between WIlliams and Orlandi, and the striker's fine header dropped into the top-left hand corner.
But any hope of that spurring on a comeback were dented somewhat just three minutes later as another Swans attack saw Dyer cross to Graham, who controlled with his first touch before drilling home from close range for his 13th goal of the season.
But as the celebrations were dying down from the home supporters, the Wolves end were up on their feet again as, amazingly, the sixth goal of the game arrived. Jarvis found himself unmarked in the box and he fired home into the bottom corner.
Wolves, buoyed by their second goal, went in search of another. And they almost got it as, following a stumble from Williams, Jarvis got into the box before unleashing a shot. But Monk slid in and timed his block perfectly.
Fletcher then worked his way into space in the box before dragging his shot wide and then Stephen Ward saw his thunderous effort palmed out by Michel Vorm.
Vorm had to be on his toes again as Monk's pass-back was too weak and the keeper had to race out and deflect the on-rushing Doyle's first touch. The striker did latch onto the loose, but the covering Leon Britton slid in to poke the ball out for a corner.
Half-time: Swans 4 Wolves 2
Brendan made a change at the break with Neil Taylor replacing Orlandi as the Swans reverted to their usual formation.
Within two minutes they went within inches of adding to their lead as first Sigurdsson saw his volley rebound out off the post before he tried his luck again only to be denied by a superb save from de Vries, who tipped his effort wide for a corner.
But while the Swans were looking to extend their lead, Wolves struck with a swift counter. Doyle slipped through for Wales midfielder David Edwards, and he slotted past Vorm and into the corner.
It set up a thrilling 35 minutes, with both sides looking for the eighth goal of the afternoon.
Fletcher flashed his header wide while at the other end Graham nodded over from Dyer's cross.
But then Wolves did get themselves level. Fletcher was the architect as his low cross skimmed along the face of goal and Jarvis sidefooted home with 20 minutes left to play.
Terry Connor then made changes with Stephen Hunt replacing Jarvis and Sylvan Ebanks-Blake went on in place of Fletcher as Wolves went in search of the winner.
Brendan made a change of his own with eight minutes left as Luke Moore replaced Graham.
Four minutes of added time were then played but neither side managed to find the winner in a truly remarkable match.