The 30-year-old joins Erik ten Hag’s side until the end of the season after United secured his exit from a temporary spell with Besiktas.
Weghorst scored nine goals in 18 appearances for the Turkish club this season, and netted twice for the Netherlands at last year’s World Cup in Qatar.
“I feel privileged to be joining Manchester United,” said Weghorst after his loan switch was confirmed late on Friday night.
“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.
“I’ve seen United’s progress under Erik ten Hag this season and can’t wait to start playing my part in pushing the team towards its aims.
“Whatever happens in the next few months, I can promise to give everything to the club for as long as I am here.
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.
“I thank everyone who has played a part in getting me to this stage, and now I am looking forward to meeting my new team-mates and getting involved straight away.”
The signing will come as a welcome boost to United, but manager Ten Hag confirmed in his press conference on Friday that the deal had not been completed before the midday registration for Saturday’s derby against Manchester City at Old Trafford.
United’s football director John Murtough said: “Wout has a consistent goalscoring record across several European leagues, and his attributes will add a further dimension to our squad for the remainder of the season.
“We were also attracted by his strong personality and dedicated approach to football, which fits perfectly with the spirit that Erik is building within our squad. He has shown a huge desire to join Manchester United and be part of what we’re striving to achieve this season.”
Besiktas said in a statement: “The temporary transfer agreement with our professional football player Wout Weghorst and his club has been terminated by mutual agreement with the player and his club.
“Due to the early termination of the contract by Burnley, a termination fee of 2,825,000 euros (£2.5million) will be paid to our company.”
Weghorst joined Burnley in a £12m deal last January after scoring 70 goals in 144 appearances for Wolfsburg.
But he found the back of the net on just two occasions for the Clarets as they were relegated to the Sky Bet Championship.
Despite Weghorst’s arrival, Ten Hag is set to be short of options in attack for Saturday’s derby with Anthony Martial a doubt.
If Martial, struggling with a leg injury, is out, Ten Hag could be forced to shift the in-form Marcus Rashford – who has scored seven goals in his last six games – in from his preferred position on the left.
Diogo Dalot is missing with a hamstring injury, while Donny van de Beek’s season is over following the knee injury he suffered in a collision with Bournemouth’s Marcos Senesi on January 3.
United were beaten 6-3 when they last faced City in October. But, since then, Ten Hag’s side have lost only once in 18 games, winning their last eight to stay alive on four fronts and climb into the top four.
Key to the turnaround has been the arrival of Brazil midfielder Casemiro.
The 30-year-old was a second-half substitute in the last derby, but he has since established himself as a key cog in United’s engine room.
“He’s so important,” Ten Hag said. “He’s the cement between the stones. His defending, his organisation, he knows his position, anticipating and pointing players into the right position, winning balls in midfield.
“He has that competitiveness to give to teams at the top level. There are not many players of this quality in the world. He is one of them and we are really happy to have him on board.”