Albion couldn't start the new year with a win but there was plenty of fighting spirit from Graham Potter's men as they shared six goals with Wolves.
Aaron Connolly scored his first Amex goal since October 2019 to give them an early lead but by half-time the visitors led 3-1 thanks to goals from Romain Saiss, Dan Burn's own goal and Ruben Neves' penalty.
But Albion were awarded a penalty 11 seconds after the restart which Neal Maupay converted before skipper Lewis Dunk headed in the equaliser with 20 minutes to go.
Adam Webster hit the bar and Leo Trossard had a goal disallowed as Albion dominated the second half but they couldn't find a fourth goal.
There had been five changes to the side which lost 1-0 to Arsenal on Tuesday. Ben White, Maupay, Trossard, Connolly and Solly March all returned while Adam Lallana was among the substitutes after recovering from a groin problem.
Joao Moutinho fired straight at Rob Sanchez and Pedro Neto shot wide of the left-hand post after showing a great burst of speed during a lively opening by Wolves, but it took just 11 minutes for Albion to score their first goal of 2021.
Adam Webster's lofted pass found Leo Trossard and he got the better of Rayan-Ait-Nouri on the right before crossing to Connolly, who lifted the ball over Rui Patricio from close range.
It was just the tonic Albion needed but frustratingly the lead lasted just seven minutes. Albion failed to clear the ball from a corner and when it was whipped back in from the left to Romain Saiss who guided a looping header past Sanchez.
Yves Bissouma picked up a fifth booking of the season – ruling him out of the trip to Manchester City on Wednesday week – before Patricio had to dive full length to his right to keep out Connolly's bicycle kick, after the Ireland striker was picked out on the edge of the box by Joel Veltman.
The right-hand side was proving a productive area for Albion with Maupay nearly making the most of a good ball in from Veltman but Wolves began to dominate and a training-ground free kick routine ended with Ruben Neves sweeping a right-foot shot just wide from the edge of the box.
It certainly wasn't the cagey affair predicted by a lot of pundits pre-kick off but Albion suffered a setback in the 33rd minute. Neto let fly from the right-hand edge of the box and Sanchez did well to push away his shot which had taken a deflection, but the ball bounced off Dan Burn's foot and rolled into an empty net.
Burn's evening went from bad to worse a minute before the break when he brought down Adama Traore on the goal-line for a stonewall penalty which Neves dispatched to make it 3-1 and leave Albion up against it.
There was a double change at the break with Andi Zeqiri and Davy Propper replacing the injured Connolly and Bissouma and Zeqiri made an immediate impact, cushioning a header into Maupay's path. The Frenchman got into the box and was brought down by Moutinho for a penalty, which Maupay got up to drill down the middle as Patricio dived to his right.
Game on and Burn had a great chance to equalise when March's cross fizzed into his path, but he couldn't generate sufficient power in his shot and Wolves scrambled the ball away. White then picked out Veltman who cushioned a header into Trossard's path but the Belgian's goalbound volley was blocked by Conor Coady.
Albion went even closer in the 67th minute when Webster headed March'ss corner against the bar and Zeqiri put the loose ball over the bar. Potter then made his final change, bringing on Adam Lallana and within two minutes Albion were deservedly level when Dunk met Trossard's corner from the left with a powerful downward header which deflected off Saiss past the helpless Patricio.
The momentum was with Albion. Referee Madley pulled up Trossard for a soft foul on Ait-Nouri before he smashed the ball past Patricio and much of the closing stages was played in Wolves' territory, but the hosts had to settle for a point and there was a tinge of relief when sub Owen Otasowie sent a free header over seconds before the final whistle.
On the spot
Neal Maupay's successful spot-kick conversion was the fifth penalty Albion have been awarded this season. Only Leicester (8) have been had more. But Albion have now conceded six penalties, the joint-highest in the league with Leeds, and only one more than Wolves.
Not Burn's night
After going 2-1 down to Dan Burn’s own goal, Albion switched to a flat back four but Burn's night went from bad to worse when he brought down Adama Traore for a penalty. He became the player to score an own goal and give away a penalty in the same Premier League game since Eliaquim Mangala for Manchester City against Hull City in September 2014.
The late-ish show
Lewis Dunk's second goal in six days was also the fourth in succession that he has scored after the 70th minute.
The line up
Sanchez, Veltman, Webster, Dunk, White, Burn (Lallana 67), March, Bissouma (Propper 45), Trossard, Connolly (Zeqiri 45), Maupay. Booked: Bissouma, Burn
Subs not used: Steele, Mac Allister, Gross, Jahanbakhsh, Bernardo, Molumby.
Referee: Andrew Madley