One of the players to produce a good performance was Facundo Pellistri, who has also told Solskjaer his preferred intention for the transfer window this summer.
Tahith Chong opened the scoring in the first-half, and then Pellistri doubled their lead, only for Colin Kazim-Richards to reduce the deficit, yet it was not enough for Derby to save themselves from losing.
Pellistri spent last season out on loan in Spain with Alaves, where he impressed, playing in 12 La Liga games, as the Spanish side narrowly escaped relegation from the top flight.
While the Uruguayan has been tipped for another loan move, he has stated his desire to stay at United and fight for his place in the squad ahead of the coming season.
After the game against Derby, the 19-year-old winger said "As you said, I went on loan to Alaves, I felt good, now I'm here trying my best and always pushing forward to the best."
"I would like to stay here but I'm still working hard, improving and will continue working."
However, United have plenty of players capable of playing on the wings, especially with Jadon Sancho set to join the club, and so it might be tough for Pellistri to find a way into the side.
Apart from Sancho, United also have the likes of Amad Diallo, Daniel James, Marcus Rashford, Mason Greenwood and Juan Mata all capable of playing on the right, and Anthony Elanga and Shola Shoretire could also be challenging for places in the squad.
Despite this, Solskjaer stated that he was impressed by Pellistri, and that a string of good performances along with getting some goals and assists could give him a boost for the coming season.
"Very good goal. Good pass, then his first touch sets him up," Solskjaer said after the game.
"Then he's very good at dribbling so going past the keeper wasn't a problem. Composed. I thought he was bright when he came on."
"I think he showed what he is because he's a difficult player to play against. One against one he can go past people so very pleased with him."
Solskjaer also claimed that Pellistri could be set for a loan deal, saying that he "can see a loan deal for him, there's been loads of interest and for his best interests a loan would be what we look for."
Pellistri signed for United from Penarol for €8.5 million last summer, and Alaves are reportedly one of the clubs interested in loaning him in once again this time around.