Solskjaer was sacked on Sunday after a disappointing performance in one win in seven Premier League The game, which included heavy defeats to Leicester, Liverpool, Manchester City and Watford.
Karik, 40, was part of the Norwegian coaching staff, but will take over as caretaker until the club works on plans to appoint an interim manager by the end of the season.
“To Ole unfortunately losing my job yesterday was hard for me and hard for a lot of people at the club,” said Carrick, who won major trophies as United player between 2006 and 2018.
“You can tell from the feelings around yesterday what Ole meant to everyone.
“We understand and ole understand that this is a results business, so as much as you build the right foundation and environment and you show people the way, you sometimes don’t get what you deserve.”
Solskjaer was also initially placed in the following temporary charge Jose MourinhoWas sacked in December 2018, but a run of 14 wins in his first 19 matches got him the job.
Carrick was fearless at the prospect of repeating that turn of events, with a focus on securing a place in the Champions League last-16 with a win against Villarreal in Spain on Tuesday.
“Whether it’s one game, two games, whether it’s a little 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 past that.
“It is a privilege to hold this position and I understand the possibilities that come with it.
“Now the focus is on changing our mindset very quickly and that is something this club has historically done very well.”
United captain Harry Maguire, who was sent off during a 4-1 humiliation at Watford on Saturday, admitted that players will have to accept responsibility for the cost of their former manager’s job.
“Players need to take responsibility for everything on and off the football field. We are the ones who cross the white line,” said the England defender.
“Ultimately, the manager paid the price and we’re all disappointed for it. Of course, we take great responsibility, we haven’t been good enough.”
Solskjaer addressed United players during a two-hour meeting on Sunday, after it was said he would be relieved of his duties.
“We fully respect what Ole has done for the club – he is a legend of the club and he will continue to be a legend at the club,” Maguire said. He said, “I don’t want to go too deep into what I said, but he wished us well.
“He said one thing that he would support us and wanted us to win every game of football, which clearly shows the man’s values.”
Maguire has been one of the main culprits of United’s terrible run as the form he showed to help England reach the Euro 2020 final left him at club level.
The world’s most expensive defender, who was one of Solskjaer’s first signings when he joined Leicester for £80m ($107m) in 2019, acknowledged things should be improving.
“We need to find something within ourselves, to find that focus, to find that mindset and to do well tomorrow and get the results that we need for this club,” he said.
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