Things looked grim for Terry Connor's side as Andrea Orlandi struck in the first minute before Joe Allen and Nathan Dyer extended Swansea's lead before the 15-minute mark.
But the visitors rallied as Steven Fletcher and Matt Jarvis pulled goals back either side of Danny Graham scoring Swansea's fourth, before second-half strikes from David Edwards and the outstanding Jarvis capped a breathless contest and secured a point.
The draw secured Swansea's Premier League status, but manager Brendan Rodgers will not have been impressed with the way his side squandered the advantage they built during an opening where Wolves' season looked set to hit a new low.
And the watching Manchester United manager Sir Alex Ferguson would hardly have seen anything to suggest Swansea will pose his side a threat at Old Trafford next weekend such was their defensive capitulation.
But anything other than a comfortable home win seemed unlikely when Wolves keeper Dorus de Vries, back at his former club, had to pick the ball out of his net twice in the opening four minutes.
Steven Caulker and Leon Britton built the play from the back for the hosts and Scott Sinclair's cross located Orlandi in acres of space for the Spaniard to find the net as his header looped over the keeper after just 25 seconds.
The hosts then doubled the lead with less than four minutes on the clock as Wolves were torn apart.
Graham and Gylfi Sigurdsson linked well to free Allen and the midfielder's shot took a deflection off Richard Stearman to leave De Vries with no chance.
Wolves had been carved open with worrying ease for the opening two goals and things rapidly got worse for Connor's side as a third arrived in the 15th minute.
Orlandi tricked his way between Kevin Foley and Michael Kightly and his deflected ball across the box was headed in at the far post by Dyer.
Graham could have made it 4-0 moments later, but he hesitated and De Vries foiled his effort to slot home from an acute angle.
Wolves managed to get a goal back when Kevin Doyle found space down the left and delivered a pinpoint cross for Fletcher to direct a free header over Michel Vorm.
But that only proved brief respite as Swansea replied with their fourth on the half hour.
Dyer was afforded far too much freedom on the right to surge into the box, and when his shot rebounded back to him, he lofted a lovely ball for Graham to coolly slot his 13th goal of the season.
The comical standard of defending continued as another Doyle cross came through to Jarvis, who cut inside Caulker to place a finish beyond Vorm three minutes later, Ashley Williams making a hash of his attempted goal-line clearance.
The lively winger could have had another soon after, but an excellent tackle from the covering Garry Monk denied him as he was about to pull the trigger.
Having been overrun towards the end of the opening half, Rodgers elected to bring on Neil Taylor for Orlandi to revert to a four-man defence.
But the relentless action continued as Sigurdsson first smashed a half volley against the foot of the post, before unleashing a left-footed strike that De Vries tipped behind for a corner.
But it was Wolves who found the net next as they sensed an extraordinary turnaround was on the cards.
Kightly's neat pass picked out the run of Edwards and the midfielder held off Monk's attentions to steer the ball past Vorm.
Fletcher then failed to get telling contact on a deep cross from Foley with the visitors fully in control and pressing for an equaliser.
And it duly arrived in the 69th minute as Fletcher's first-time ball across the six-yard box was slid home by Jarvis for the winger's second of the game.
All the momentum was with Wolves and Kightly should have given Connor a first win in his 11th game in charge, but blazed high and wide.
At the other end De Vries did enough to deny substitute Luke Moore, while referee Jon Moss waved away handball appeals against Stearman in a breathless, frantic finale.