The Toffees' patchy results this season have seen the Portuguese's position come under intense scrutiny and he will not be sitting any more comfortably after Wolves picked up a third straight Premier League victory to cement their place just outside the top six.
Ruben Neves' early penalty had Nuno Espirito Santo's side ahead but Everton were level midway through the first half thanks to Andre Gomes' thumping first goal for the club.
However, slack marking from a free-kick allowed Raul Jimenez to head home on the stroke of half-time and Leander Dendoncker smashed in a volley midway through the second period to secure Wolves' first win at Goodison in 40 years.
Everton had taken the lead inside three minutes of their win at Huddersfield Town on Tuesday but it was their turn to fall behind early on here as Leighton Baines felled Matt Doherty and Neves calmly converted the spot-kick.
The hosts almost gifted Wolves a second in the 20th minute when Tom Davies lost the ball and Jimenez fed Dendoncker, who was denied by Jordan Pickford at close quarters.
Rui Patricio surged out to thwart Theo Walcott as Everton sought a response, which duly arrived in the 27th minute thanks to Gomes' rasping drive into the roof of the net from 18 yards.
But Wolves restored their advantage as Jimenez rose unchallenged to head Joao Moutinho's delivery into the bottom-left corner.
And the visitors earned some breathing space as the rebound from Diogo Jota's blocked attempt fell straight to Dendoncker to lash home from eight yards - capping an impressive day for Wolves and a torrid one for Silva and Everton.
What does it mean? Wolves the best of the rest
Wins over Leicester City, West Ham and now Everton have seen Wolves rise to seventh in the table. The usual suspects make up the top six, which has proven almost impossible to break into, but there can be no doubt that Nuno's men offer a tough test for anyone in this division.
A cat-astrophic day for Everton, Wolves 'feline' good
Amid a miserable afternoon for Everton fans, there was some light relief in the aftermath of Wolves' third goal as a black cat strayed onto the pitch and delayed proceedings for a couple of minutes before the intruder was eventually chased away by a steward.
Toffees come unstuck at the back once more
Everton's shaky defence has been cause for concern for most of the season and they contributed to their own downfall again as Baines' clumsy foul led to the first goal and no one picked up Jimenez for the second.
Key Opta Facts
- Wolves have won 11 Premier League games this season – their joint-highest tally in a single Premier League campaign, alongside the 2010-11 season.
- Wolves have scored 3-plus goals in three consecutive top-flight matches for the first time since March 1980.
- Everton have lost three of their last four Premier League games at Goodison Park, as many as they had lost in their previous 23.
- Jimenez has been directly involved in more Premier League goals than any other Wolves player this season (9 goals, 5 assists).
It does not get any easier for Everton, who host champions Manchester City on Wednesday. Wolves have an FA Cup fourth-round replay at home to Shrewsbury Town on Tuesday before returning to league action against Newcastle United a week on Monday.