Carrick fills the vacancy left by Ruben Amorim and takes over from Darren Fletcher, who was placed in temporary charge of first-team duties.
Former players Carrick and Solskjaer were interviewed for the interim role by United's chiefs Omar Berrada and Jason Wilcox, who had been leading the search.
Despite Solskjaer's popularity at Old Trafford, United leaned in Carrick's direction, having been impressed with his plans for the rest of term.
"Having the responsibility to lead Manchester United is an honour," Carrick told the club's website.
"I know what it takes to succeed here; my focus is now on helping the players to reach the standards that we expect at this incredible club, which we know that this group is more than capable of producing.
"I have worked with a number of the players already and have obviously continued to watch the team closely in recent years, I have total belief in their talents, dedication and ability to be successful here.
"There is still a lot to fight for this season, we are ready to pull everyone together and give the fans the performances that their loyal support deserves."
Carrick previously managed the club as a caretaker for three games following Solskjaer's dismissal in November 2021, before stepping down for Ralf Rangnick's arrival.
He was appointed the boss of a then-struggling Middlesbrough in October 2022 and guided them to the Championship play-offs in his first season.
But Boro missed out on the play-offs in the next two campaigns and Carrick was sacked at the end of the 2024/25 season after 136 games in charge.
Carrick made 464 appearances for United as a player, winning 12 major trophies in as many seasons.
United host rivals Manchester City on Saturday in what will be Carrick's first game in charge.