Fabio Silva's opener on the stroke of half-time, which looped over Sam Johnstone after the 18-year-old had blocked Kyle Bartley's clearance, was cancelled out by Mbaye Diagne's header just after the hour mark.
The point keeps Wolves in 12th place on 42 points, while Albion will now need to win all of their remaining games to have any hope of survival.
Youngsters impress for new-look Wolves
Interestingly, all three players introduced by the Portuguese were 21 or under, and all impressed. Fabio Silva's luck in front of goal seems to have turned which was epitomised by his flukey opener, while Owen Otasowie provided a presence and physicality that Wolves' midfield hasn't seen for a while.
However, the most impressive performance in old gold came from Vitinha, who the Molineux faithful have been crying out for on social media for months. The 21-year-old, who was making just his third Premier League start, looked comfortable on the ball and was involved in some brilliant intricate build-up play for Silva's opener.
With Wanderers' Premier League status officially secured for 2021-22 this weekend, fans will be hoping that these three youngsters will see plenty more action in the remaining four games of the campaign.
Diagne header emphasises need for defensive reinforcements
Wolves' attacking performance probably warranted three points at the Hawthorns since 1996, a defensive relapse let them down for the umpteenth time this season.
Two saves from Rui Patricio early in the second half to deny Conors Townsend and Gallagher kept Wolves' lead intact, but the 33-year-old was let down by his backline just after the hour mark when Mbaye Diagne was allowed a free header from six yards out, and the Senegal international made no mistakes.
After this, last week's horror show and the 3-2 defeat to West Ham at the start of April, every game that goes by seems to reemphasise the need for Wolves to buy at least two first-team centre backs in the summer.
Conor Coady and Romain Saiss have struggled in a back four all campaign, while Willy Boly's injury struggles and advancing age means he can't be relied upon as much as he used to be.
After three years of neglecting the heart of their defence, the time has come for Jeff Shi and the board to spend big in the summer.
Is Nuno the man to transform Wolves' style?
While failure to beat their local rivals will not please Wolves fans, the performance against West Brom was much improved when compared to recent outings.
Wanderers were enjoyable to watch and created 20 chances, 10 of which were on target, their highest number since the reverse fixture against the Baggies.
Despite putting in some incredibly dull performances this campaign and scoring just 33 goals in 34 Premier League games, Nuno has proven that his team can play attractive football when it wants to. With Raul Jimenez back next season, there should be no excuse for Wolves not to enjoy a much more entertaining and successful campaign.
Whether this will happen, or become a long-term plan under the Portuguese, remains to be seen.