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Rangnick steps in to replace Solskjaer, who was removed after United's 4-1 defeat by Watford this month.
Former player Michael Carrick, a member of Solskjaer's coaching staff, will take charge of the United team for Tuesday's Champions League match away to Villarreal.
The 21-year-old England international has started only two of Manchester United's five Premier League games since his move from Borussia Dortmund.
The defender now becomes the third English player to ever play for Inter, following in the footsteps of Gerry Hitchens and Paul Ince.
Thirteen months on from the match that finally drew the curtain on Jose Mourinho's time in charge of Manchester United, the Red Devils again find themselves living in Liverpool's shadow ahead of their return to Anfield on Sunday.
The final is on March 1 when City will seek to win the trophy for a third successive year and a fourth time in five seasons.
Solskjaer's problems have mounted amid claims that United midfielder Paul Pogba wants to join Real Madrid, while the boss is reportedly unhappy with the fitness levels and commitment of some of his squad.
Solskjaer, who took interim charge of the club in December, won 10 of his first 11 games in charge and was handed the job on a permanent basis in March only for results to nosedive in the months that followed.
It was yet another Premier League game with VAR controversy which will continue to provoke debate about the system introduced this season.
After a 4-0 home win over Chelsea on the opening day, Ole Gunnar Solskjaer's team dominated the early exchanges and led after 27 minutes when Marcus Rashford slipped the ball in for Martial to score his 50th goal for the club in all competitions.
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