Carrick will lead United's coaching staff for Tuesday's Champions League trip to Villarreal, with a place in the knock-out stages up for grabs.
The former United midfielder and assistant coach to Solskjaer is acting as a caretaker while United search for an interim candidate to take charge until the end of the season.
United are also seeking Solskjaer's permanent successor for next season, with Mauricio Pochettino understood to be interested in swapping Paris Saint-Germain for the Old Trafford dugout.
Solskjaer's remarkable start as an interim manager eventually convinced United to appoint him permanently. Carrick is focused on the trip to Villarreal, but is ready to be in charge for however long is necessary.
"My thought process is picking the team for tomorrow night, giving it my best. I know the boys very well, the staff very well, the club very well," he said, having joined as a player from Tottenham Hotspur in 2006.
"This club's been my life for an awful long time. It's the sacrifice and dedication you've got to get and you've got to give.
"I'm prepared to do it for however long it takes, whether it's one game, two games, whether it's a little stretch longer than that, right now I'm not too sure, but that's not something that's on my mind. It’s about tomorrow.
“First and foremost, it's literally been a day, just over 24 hours, since everything unfolded, and the game is not much further away in terms of time, so that is all I've been thinking about, the game tomorrow night. Not about today, yesterday. I've not looked past that.
“Yes of course we've got games coming up after that, a big game at the weekend, but we'll address that when we need to, and however long the club want me to be here.
“I'll do my best and whatever results we get – hopefully we get good ones – and we should get good ones, I truly believe that, and we'll wait and see what happens after that.”
Carrick has been tasked with preparing United for an important Champions League game at just 24 hours notice, having learned of Solskjaer's dismissal and his promotion to caretaker on Sunday.
"I found out yesterday morning at the training ground. Didn't know anything before that point," he said.
"I arrived and spoke to Ed. It was bit of a rushed end to organising training and the day's work. Everyone adapted, reacted, dealt with it in a real classy manner.
"It’s a challenge, the initial reaction yesterday and coming to terms with the situation. But quite quickly you’ve got to focus, there’s a responsibility.
"I'll just throw myself into doing everything I can, of course it’s a limited time, I’m well aware of that, it's a challenge I’m relishing at the moment, I’m really looking forward to it.
"I’ve got great people working with me, a great group of players, they’ve proved that before and they’ll prove that again."
Carrick will be assisted by Mike Phelan and Kieran McKenna, who have also stayed on despite Solskjaer's departure, and suggested that he will put his own personality
"We'll have to wait and see. I've obviously worked closely with Ole for a long time now and we have very similar beliefs, we did as players and we do as coaches and managers.
“I have my own personality compared to Ole but we worked together well. I'm not giving too much away in terms of plans but we're clear in what we want to do and we're looking forward to seeing it on the pitch.”
Register for free to continue reading
Registration is a free and easy way to support our truly independent journalism
By registering, you will also enjoy limited access to Premium articles, exclusive newsletters, commenting, and virtual events with our leading journalists
Already have an account? sign in
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies