United manager Erik ten Hag was determined to bolster his attack following Ronaldo's exit in November.
Weghorst, who was already on loan at Turkish club Besiktas, will not be eligible to play against Manchester City in the English Premier League on Saturday because he was not registered in time.
"Wout has a consistent goalscoring record across several European leagues, and his attributes will add a further dimension to our squad," United football director John Murtough said.
Weghorst appears an unlikely replacement for Ronaldo, who had his contract terminated by United in November after an explosive interview in which he criticised Ten Hag and the club owners.
Ronaldo has since signed for Saudi Arabian club Al Nassr.
"I feel privileged to be joining Manchester United," Weghorst said. "I've played against the club in the past and it's a fantastic feeling to now have the chance to pull on the famous red shirt.Â
"I've seen United's progress under Erik ten Hag and can't wait to start playing my part in pushing the team towards its aims."
Ten Hag was keen to bring in cover in attack to boost his team's push for Champions League qualification and its pursuit of a trophy in his first season in charge.
Ronaldo's departure left United with only two senior strikers in Anthony Martial and Marcus Rashford.
Weghorst played for the Netherlands at the World Cup in Qatar and scored twice in the dramatic quarte-rfinal against Argentina. His goals sent the match to extra time and penalties before Argentina won 4-3 in the shootout.
The 30-year-old Weghorst has eight goals for Besiktas this season, including two in his last three games.
Weghorst joined Burnley from German club Wolfsburg in January last year. He managed only two goals in 20 appearances as Burnley was relegated from the Premier League.
United were linked with a move for Cody Gakpo this month but the Netherlands forward instead joined rival Liverpool.
Ten Hag is expected to move for a long-term replacement for Ronaldo in the summer, with Harry Kane among the players linked.
United midfielder Donny van de Beek has been ruled out for the rest of the season by a knee injury.
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