Pogba suffered global lambasting after missing from the spot-kick, with some section of the United faithful racially abusing the former Juventus player.
But Solskjaer has revealed the 26-year-old has overcome the criticisms he faced after the game, declaring his readiness for the subsequent games.
"Paul's fine, Paul's a strong character and it makes him stronger," Solskjaer said, speaking ahead of Saturday's match against Crystal Palace.
"When you speak to him he's fine and I just cannot believe we're still sitting here in 2019 talking about these instances.
"Social media is a place where people can hide behind fake identities. It's not down to me to change it but there are so many Ole Gunnar Solskjaers on Twitter, and I know it's not me.
"We've got to do something about it and the authorities have got to do something about it.
"You feel sorry for [the social media trolls], they must have problems themselves.
"We need to protect individuals and protect people. When there are death threats and racism, it's serious allegations, serious incidents.
"Sometimes you get annoyed, what can we do about it as Manchester United? We're not going to ban our players from using social media. We have to spread the good word instead.
"We can use it to spread good and have to stop those incidents."
Having stressed on the readiness of Paul Pogba for the remaining games, Solskjaer also lamented on the penalty duties for his side.
"It's not that I've left it to the players, we've nominated two and throughout the season you'll have games where they are not playing and it's good to know they're there and trusted to take penalties," the Red Devils boss said.
"It's not like anarchy and they can do whatever they want, we have a few, last year we had Marcus, Jesse [Lingard] and Paul.
"It's not left to them"
