The club condemned the number of abusive messages sent to the pair on social media after the 2-1 defeat on Wednesday and the United manager said those responsible were not welcome to support the club.
- Stream ESPN FC Daily on ESPN+ (U.S. only)
- Insider Notebook: Man United concern over Rashford
- Dawson: Fernandes draws comparisons with Cantona in first year
"It's just incredible we have these scenes still, this abuse in 2021," Solskjaer told a news conference on Friday.
"We've been campaigning for a long time now with the Premier League and I think it's working but there are still some people that haven't got it.
"They hide behind social media and be anonymous. It's unacceptable and it's disgusting. They're not Man United fans when this happens. We're all standing behind every single one of our players.
"Axel is fine, I've spoken to him, and of course it's not nice but he's a strong boy, strong character."
Premier League Chief Executive Richard Masters released a statement on Friday condemning the abuse. It read: "I am appalled to see the racial abuse received by players this week. Racist behaviour of any form is unacceptable and nobody should have to deal with it. Tackling online hate is a priority for football, and I believe social media companies need to do more.
"We at the Premier League will continue to support players, managers and their families who receive discriminatory online abuse. Through our central reporting system we have been able to take immediate action on cases like this and we investigate every report received, with legal action taken where possible.
"We are in regular dialogue with social media companies, challenging them to do more against discriminatory abuse on their platforms. We want to see swifter removal of offensive messages and improved identification and banning of offenders."
The 28-year-old is set to move to the London Stadium until the end of the season and Solskjaer said he expects the midfielder to return to the club "revitalised."
"We want Jesse to come back here revitalised and having shown at West Ham how good a player he is," Solskjaer added.
"He's just been unfortunate with a couple of issues, some periods he's had to stay away from us with COVID -- isolation and close contact.
"The rest of the team has done well, we've had little to no injuries and he's not forced his way into the team. We want him to come back and show he's a Man United player through and through."
Solskjaer is also expecting Marcos Rojo to join Boca Juniors before the end of the January transfer window but goalkeeper Sergio Romero could remain at the club until the summer despite not featuring at all this season.
"It's likely that Marcos goes back to Argentina, there's little bit of paperwork left to be done," Solskjaer said. "With Sergio, I've not heard anything there apart from that he'll be with us for the rest of his contract."
United travel to Arsenal on Saturday looking to bounce back from the shock home defeat to Sheffield United on Wednesday.
Solskjaer has defender Eric Bailly available after being sidelined in midweek through injury.
"He's travelling to Arsenal," Solskjaer added. "Apart from that we didn't get any other losses from Sheffield United and the players are all fit and raring to go."