Solskjaer, who signed a new contract this summer to keep him at Old Trafford until 2024, has been under fire after a string of disappointing results.
An early exit in the EFL Cup and unconvincing performances in both the Premier League and Champions League have raised concerns about the manager.
Asked after Tuesday's 3-2 Champions League comeback triumph over Atalanta if Solskjaer has the players' backing, Fernandes told Brazil's TNT Sports: "Of course, obviously, he is our coach. We follow him every day. We show that. We follow his ideas and we will continue to do so.
"We believe in our coach and our staff... we need to do what the coach tells us because our coach is the maximum exponent of what we must respect. He has been here for three years and we've done good things. It's true that we haven't won anything up to now, but the team has shown it has grown a lot.
"We have a lot to improve on and the coach also knows there has to be some improvement at their end but that is part of football. Every day we are learning something new. But football is the present and it's the now."
Fernandes highlighted his team's determination to turn around what looked to be a disastrous night at Old Trafford.
United had gone into the game having won only one of their previous five games in all competitions and were 2-0 down after 28 minutes played.
The Red Devils scored three second-half goals, with Cristiano Ronaldo heading in the winner, to go top of Group F.
"It was the team's will, the will to win, to work as a team, to work all for each other that brings these results," the Portugal international added. "We have to continue like this. Technically and tactically, it may not have been our best game. We made mistakes and we have to improve. Just because we've won all is not right but by the same token, just because we lost the other day [at Leicester City], all was not bad.
"We have to bring out the good things we did against Atalanta and grow from there. We have to believe in ourselves, try and not give up and more than anything, know that we are here for each other."
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Fernandes continued: "Manchester United always has to think of having a starring role every season. It's true that this club demands to win games, to win titles...We have a big game coming up, we have a game that we need to win, an important one against one of the biggest rivals of this club.
"We need to get something from that game. We must do it. We have taken one point in three league games and it's not enough. We need to prove that we are here, that we will be in the fight but to be in the contention we have to win game after a game and the next one is against Liverpool."
Solskjaer, meanwhile, believes the fans' support will again be crucial.
"In the second half [against Atalanta], together with the crowd, the players were incredible," the Norwegian said. "Attitude, work rate, quality, everything a United team should have. It's what Manchester United do, it's in our DNA. We tend to make it hard on ourselves and we did."
Asked if he thought that he should listen more to the noise from the atmosphere at Old Trafford than from the noise of social media, he said laughing: "Yes! I think so. Definitely. Our fans are the best in the world. But there's always going to be some criticism when you don't win every game. That's expectations at United. But we have to keep the noise away, enjoy the atmosphere here at Old Trafford] and believe in ourselves."