Matheus Pereira scored two penalties in each half as the Baggies came back from 2-1 down to defeat win the first Black Country derby since 2012.
Story of the match
Persisting with the back four, the hosts' had the games opening attempt on target when Boly's left-footed hook forced David Button into an early save.
Wolves' defensive woes continued as Michael Oliver rewarded West Brom a penalty after Boly brought Callum Robinson down. Despite the tight call, the decision stood and Pereira tucked the penalty away to put the visitors ahead.
Going a goal behind did kick Wolves into gear, with Adama Traore and Leander Dendoncker testing Button from range.
However Wolves made their dominance pay in the final ten minutes after the Baggies failed to clear the ball from a free-kick, with Silva equalising from close range.
The ball bounced around in the penalty area, before the young striker neatly tucked the ball into the back of the net, his first goal at Molineux.
Moments later, they had turned the game around through Boly's first goal since his injury-time winner in Besiktas last season. Joao Moutinho's dangerous corner wasn't cleared by the Baggies, and Boly was the quickest to react, stabbing the ball past Button.
With the defender scoring, grabbing an assist and conceding a penalty, he is the first Wanderers player to do so since Adlene Guedioura in the same fixture the last time Wolves beat their rivals.
Nuno Espirito Santo's side, who came into the game without a league win in five, started the second half the brighter, with Silva and Dendoncker trying their luck.
But it was West Brom who scored next, against the run of play it has to be said. A long throw was flicked on by Kyle Bartley onto the head of Ajeyi, whose looping header beat Rui Patricio.
Ajeyi's third goal of 2021 kicked Allardyce's team into gear and they restored their early lead through another controversial penalty after Conor Coady was deemed to have fouled Robinson.
Wanderers' best chance to equalise fell to Cutrone moments after coming on, but the Italian striker fired the ball into the ground and over despite being left unmarked.
As the game entered it's final moments, Wolves threw everything they could at West Brom, with both Cutrone and Silva on the pitch for the first time.
Takeaway from the match
Is it time for Wolves to revert back to what they know?
Nuno has opted for a back four in the previous four games, winning one, losing two and drawing one. Their last league result came after a shift in formation to the back three saw them come from behind to beat Chelsea 2-1.
Labelled as a transitional year at the club, the shift in style has saw mixed performances as a more possession-based system being implemented.
However, with Wolves now only 10 points clear from the relegation spaces and distancing themselves from European football, going back to what secured them back-to-back seventh-placed finishes could be on the cards.