Southampton, currently at the bottom of the Premier League, nearly pulled off an upset at Old Trafford on Thursday, but Amad Diallo's remarkable hat-trick in just 12 minutes turned the match in United's favor.
This victory came on the heels of a challenging 2-2 draw with Liverpool and a confidence-boosting FA Cup win against Arsenal.
Amorim, who succeeded the dismissed Erik ten Hag in November, had previously cautioned that a "storm will come" as the team faced a challenging stretch, resulting in six defeats in eight matches.
The 39-year-old Portuguese manager anticipates further challenges after a lackluster performance against the struggling Saints.
Despite the win, United remains in 12th place in the Premier League standings as they prepare to host Brighton on Sunday.
“I think it’s going to be hard until the end of the season,” Amorim said. “It’s going to be a little bit that rollercoaster. Then it depends on the time you have to train. We are still in the (FA) Cup and we have UEFA games for the Europa League.
“Then it’s important also to focus on next Thursday (when Scottish giants Rangers visit Old Trafford) to win and to try to be in that top eight.”
He added: “We had seven trainings together in 14, 15 games. It’s really hard, but we have to continue and try to win it. Sometimes playing good, sometimes playing bad.”
Manager Amorim acknowledged his team’s subpar performance against Southampton, citing fatigue among several players, notably Manuel Ugarte, whose late first-half own goal proved decisive for the visiting team.
“You (the media) are focused on what I said about Ugarte,” he said.
“Ugarte was tired because he did a great job against Liverpool.
“If you look at the game against Arsenal, maybe Ugarte and Bruno (Fernandes) were the best players against Liverpool.
“They didn’t play against Newcastle. I think that happens with all players in the Premier League.”