After exiting the League Cup to fourth-tier Grimsby on Wednesday, the Red Devils twice blew the lead at Old Trafford on Saturday.
But they were saved in stoppage time when Fernandes, who missed a penalty in last weekend's 1-1 draw at Fulham, scored from the spot after a VAR review for a pull on Amad Diallo.
Victory takes United onto four points from their opening three Premier League games and eases the pressure on Amorim ahead of a two-week international break.
"I'm not thinking about turning points," said Amorim after securing just his eighth win in 30 Premier League games in charge.
"I had this conversation with you guys (the media) like 10 times, so it's day by day...we have a lot to do, but we returned a little bit to our level today."
The 20-time English champions had suffered the latest and biggest embarrassment of Amorim's reign in losing to League Two opposition for the first time in the club's history in midweek.
The Portuguese coach cast doubt on his future in the aftermath, saying "something has to change" and that he would "think things through".
'Hard' for Man Utd 'keepers
Amorim dropped goalkeeper Andre Onana after his role in both Grimsby goals, but his replacement Altay Bayindir endured another difficult afternoon.
United have been linked with a move for Antwerp's Senne Lammens as a solution to their goalkeeping issues before Monday's transfer deadline.
"It's hard to be a Manchester United goalkeeper in this moment. But if you look at the first goal, we can defend the better the box," added Amorim.
"The players are struggling a little bit with all the things around the club, that is normal. So it's not just the goalkeepers, I think everybody has to improve."