Bruno Fernandes scored twice and Edinson Cavani grabbed his first for the club.
Manchester 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 UEFA Champions League.
But Manchester United, who move up to 13th and 10 points, responded with a display which, while far from smooth, was full of character.
This was epitomised by captain Harry Maguire’s determined display at the back.
“We knew we had to get three points today. It’s early in the season and we don’t look at the table, but the points we have accumulated isn’t good enough for this club,” said Maguire.
“It’s important to bounce back from setbacks. We’ve been questioned but we know we don’t answer them in 90 minutes, we answer them day in, day out.
“The lads are working tirelessly for this club. Everyone wants to improve and do better. The manager has to pull the lads off the training ground.”
The pressure on Solskjaer intensified when Everton took the lead in the 19th minute with Brazilian winger Bernard’s low drive squeezing through Aaron Wan-Bissaka’s legs into the bottom corner.
Six minutes later though, Manchester 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 Manchester 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.
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.
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 problem was that after we scored we were not able to defend well, and they took advantage. They were able to score easily.
“We said before the game we had to defend better and we did not defend good enough. We were slow at the back, we have to improve,” said the Italian.
“I know how to manage difficulties, every year you have them. We can start again after the international break,” added Ancelotti.(Reuters/NAN)