The club have announced the Norwegian has sealed a three-year contract, keeping him in the hot seat until 2022.
Executive vice-chairman Woodward led the extensive process to find Jose Mourinho’s permanent successor and was due to wait until the summer.
But Solskjaer’s incredible three-month spell as interim boss has convinced United to confirm his role now with six weeks of the season to go.
Woodward suggests the reasons for Solskjaer’s appointment are his top results of 14 wins from 19 games since taking charge, the impressive performances, his huge experience, his insistence on giving youth chances and the fact he is a favourite with the Old Trafford fans.
“Since coming in as caretaker manager in December, the results Ole has delivered speak for themselves,” Woodward told United’s official website.
“More than just performances and results, Ole brings a wealth of experience, both as a player and as a coach, coupled with a desire to give young players their chance and a deep understanding of the culture of the club.
“This all means that he is the right person to take Manchester United forward.
“I want to thank Ole and the coaching team for everything they have done so far and congratulate him on this richly deserved appointment.
“The fans and everyone at the club are behind him as he looks to take us where we need to be and build the next stage of our history.”
Solskjaer will speak to the media for the first time since the announcement tomorrow morning as he conducts his pre-match press conference ahead of Watford’s visit to Old Trafford on Saturday (3pm).
The Premier League clash will be crucial in fifth-placed United’s pursuit of a top-four finish, which looked almost impossible when Jose Mourinho was axed in mid-December.
Solskjaer, though, has transformed the club on and off the pitch, with United now just three points behind third-placed Tottenham and through to the Champions League quarter-finals where they will play Barcelona in April.