Captain Lewis Dunk nodded in Albion's 70th-minute equaliser after Neal Maupay's penalty early in the second half sparked a seemingly unlikely comeback.
Wolves had complete control of the contest at half-time thanks to Romain Saiss' header, a Dan Burn own goal and Ruben Neves' spot-kick following Aaron Connolly's early opener.
Nuno Espirito Santo's visitors crumbled after the restart and may be thankful not to leave the south coast empty handed as the Seagulls also struck the crossbar through Adam Webster during a vastly-improved second-half showing.
Graham Potter's beleaguered hosts remain without a home top-flight win this term but showed plenty of the fighting spirit they will need to avoid relegation.
The Seagulls move three points clear of the bottom three, albeit having played two games more than 18th-placed Fulham, while Wolves remain 12th.
Albion came into the new year on the back of a dismal 2020 which brought just five Premier League wins, including a solitary success on home soil.
Head coach Potter brought back attacking trio Leandro Trossard, Maupay and Connolly as part of five changes in search of only a second success in 15 league outings.
The alterations paid almost immediate dividends to give the hosts a 13th-minute lead.
Trossard twisted and turned Rayan Ait-Nouri on the right flank before delivering an in-swinging cross for Connolly to arrive ahead of Wolves goalkeeper Rui Patricio and poke home.
The breakthrough goal was Connolly's first strike on home soil in 15 months but things swiftly went downhill.
Morocco international Saiss levelled just six minutes later, diverting a looping header beyond Robert Sanchez following Nelson Semedo's right-wing cross to score for the second time in three games.
Wolves manager Nuno was this week charged with improper conduct by the Football Association over comments he made about referee Lee Mason following last month's defeat at Burnley.
His dominant side deservedly edged ahead, albeit in fortuitous fashion, 11 minutes before the break.
Pedro Neto's stinging effort from the right deflected off Solly March and was repelled by Sanchez, only for the ball to rebound off the unfortunate Burn and trickle over the line.
Burn's miserable evening continued as he conceded a penalty a minute before the break.
After being beaten for pace by Adama Traore on the Wolves right, he brought down his opponent with a mistimed lunge.
Midfielder Neves took full advantage of the situation, coolly sending Sanchez the wrong way from 12 yards to claim his first goal of the season and increase the difficulty of Potter's half-time team talk.
Based on the opening 45 minutes, a fightback looked fanciful. Yet the Seagulls were given immediate cause for optimism as they halved the deficit within a minute of the resumption.
Maupay, who missed from the spot in the 1-1 draw with Liverpool last month, drew a foul from Joao Moutinho inside the box before picking himself to slam the ball down the centre of the goal.
Albion almost levelled in the 68th minute but Webster's header from a March corner bounced back off the woodwork.
Wolves did not learn from that let off and were again susceptible at set-pieces as Dunk equalised.
The centre-back found space at the far past to powerfully connect with Trossard's right-wing corner and nod beyond Patricio.
Albion had the better of the final 20 minutes without creating much and it was Wolves who went closest to a winner.
Unmarked substitute Owen Otasowie was presented with a golden chance deep into added time but somehow headed over at the end of a pulsating match.
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