United have failed to win at home in the league this season and there were questions over Solskjaer's future after the midweek defeat to Istanbul Basaksehir in the Champions League.
The loss was the third straight defeat for Carlo Ancelotti's side who, after being the early season pace-setters, have now slipped to fifth in the standings.
The pressure on Solskjaer intensified when Everton took the lead in the 19th minute with Brazilian winger Bernard's low drive through Aaron Wan-Bissaka's legs squeezing in the bottom corner.
UEFA suspends official involved in PSG-Basaksehir player walk-off
Six minutes later though, United drew level when Fernandes met a Luke Shaw cross from the left with a perfectly angled and firm header.
Everton left back Lucas Digne then struck the post from a promising position but United were growing into the game and it was no surprise when they took the lead.
Fernandes floated in a cross from the left which Marcus Rashford rose to head into the bottom corner but although the striker failed to connect, the ball went in off the post.
Substitute Cavani added the third in stoppage time, finishing off a counter-attack led by Fernandes for his first goal since joining the club on deadline day last month. (Reporting by Simon Evans Editing by Christian Radnedge)
Barca to steal Bayern target from Real as Laporta returns - Euro Papers
'There is more to come' - Gerrard says Rangers still have work to do