Solskjaer was sacked on Sunday after a dismal run of one win in seven Premier League games, which included heavy defeats to Leicester, Liverpool, Manchester City and Watford.
Carrick, 40, was part of the Norwegian's coaching staff, but will take caretaker charge while the club work on plans to appoint an interim manager until the end of the season.
"To see Ole unfortunately lose his job yesterday was tough for me and tough for a lot of people at the club," said Carrick, who won a clutch of major trophies as a United player between 2006 and 2018.
"You could tell by the emotion around the place yesterday what Ole meant to everyone.
"We understand and Ole understands it's a results business so as much as you build the right foundations and environment and you show people the way, you sometimes don't get what you deserve."
Solskjaer was also initially put in temporary charge following Jose Mourinho's sacking in December 2018, but a run of 14 wins in his first 19 games earned him the job.
Carrick was coy on the possibility of replicating that turn of events, with his focus on securing a place in the Champions League last-16 with victory against Villarreal in Spain on Tuesday.
"Whether it's one game, two games, whether it's a little stretch longer than that, right now, it's literally about tomorrow," he said. "It's a big game, obviously for us in terms of the group stages and trying to get through.
"It's a privilege for me to sit in this position and I understand the possibilities that come with that.
"The focus now is on flipping our mindset very quickly and that's something this club has done very well historically."