The Old Trafford hierarchy sacked Mourinho in the aftermath of Sunday's 3-1 defeat at Anfield that left the 20-times champions an embarrassing 19 points behind league leaders Liverpool after just 17 matches.
Former United striker Solskjaer, the current manager of Norwegian side Molde, will take charge of the first team with immediate effect and will remain in place as caretaker manager while the club looks for a new full-time boss.
Tottenham manager Mauricio Pochettino and former Real Madrid boss Zinedine Zidane are among the bookmakers' favourites to take the United job on a full-time basis at the end of the season.
Mourinho was axed on Tuesday after just two-and-half years in charge at Old Trafford, with United in sixth place in the Premier League and facing an uphill task to qualify for next year's Champions League
Alex Ferguson's former right-hand man Mike Phelan will join the 45-year-old Solskjaer as first-team coach, working alongside Michael Carrick and Kieran McKenna.
"Manchester United is in my heart and it's brilliant to be coming back in this role," said the Norwegian. "I'm really looking forward to working with the very talented squad we have, the staff and everyone at the club,"
Executive vice-chairman Ed Woodward said: "Ole is a club legend with huge experience, both on the pitch and in coaching roles.
"His history at Manchester United means he lives and breathes the culture here and everyone at the club is delighted to have him and Mike Phelan back. We are confident they will unite the players and the fans as we head into the second half of the season."
David Beckham sent his former teammate a message of support on Instagram that read: "Good luck Ole".
Norwegian Prime Minister Erna Solberg tweeted: "FINALLY CONFIRMED! Great day for Norwegian football. Good luck keeping control of the Red Devils, @olegs26_ole."
United are hoping that Solskjaer, who coached Paul Pogba and Jesse Lingard in the reserves, will inject renewed positivity into an ailing side, whose underperforming players and drab style played a big part in Mourinho's exit.
Solskjaer is seen as a safe pair of hands who understands the club, as United target a return to the pinnacle of English football -- a position they have ceded to bitter rivals Manchester City since Ferguson's retirement in 2013.
The club are understood to be looking to hire a head of football, reporting to Woodward, to oversee football operations -- a role it is believed Mourinho had a long-standing reluctance about.
Solskjaer joined United as a player from Molde in 1996 and was part of Alex Ferguson's team that won the treble of the Champions League, Premier League and FA Cup in 1998/99.
He scored 126 goals in 366 appearances for United between 1996 and 2007, winning six Premier League titles, and in 2008 he became the club's reserve-team manager, before taking the managerial reins at Norwegian club Molde.
He won the league title for the first time in Molde's history in 2011 and repeated the feat the following year.
The Norwegian's first game in charge of Manchester United will be Saturday's trip to former club Cardiff.
Solskjaer managed the Bluebirds between January 2014 and September 2014, failing to stop them being relegated from the Premier League, returning to Molde in 2015.
Speaking moments before Solskjaer's appointment was made official, former United first-team coach Rene Meulensteen warned it would be a huge job for the Norwegian.
"It would be a good choice in terms of the fact that they want to bring back some of the spirit that was at United in the time of Sir Alex Ferguson, and Ole Gunnar Solskjaer was a big part of that," he told Sky Sports News.
"In that respect I can understand they were looking for a candidate who can live up to those expectations."
Molde have confirmed that Solskjaer will return to the club in May, with chief executive officer Oystein Neerland saying they are happy to "lend" their coach to United.
The Manchester United Supporters' Trust issued a statement backing the appointment of Solskjaer.
"United need a lift and our ultimate super-sub, Ole Gunnar Solskjaer, is just the man for the job," MUST said in a statement.