The Netherlands international has cost United £2.6m, with the Championship leaders in turn paying that fee to Besiktas after the Turkish club agreed to cancel his loan in Istanbul.
Weghorst will spend the rest of the season at Old Trafford but United, who will pay 100 per cent of his wages, have no option or obligation to buy him in the summer.
Manager Erik ten Hag had prioritised a centre-forward in the January transfer market after Cristiano Ronaldo left United and, with their budget restricted after they overspent last summer, they turned to the loan market.
United believe the 6ft 6in Weghorst, who signed too late to be registered in time to play in Saturday’s Manchester derby, offers a different dimension to their other striking options, in Anthony Martial and Marcus Rashford, and were impressed by his fine goalscoring record over his career.
The 30-year-old forward said: “I feel privileged to be joining Manchester United. 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 thank everyone who has played a part in getting me to this stage, and now I am looking forward to meeting my new teammates and getting involved straight away.”
Weghorst’s only taste of the Premier League brought just two goals in 20 matches for Burnley as they were relegated after his January arrival last year but he averaged a goal every two games for Wolfsburg in the Bundesliga, with 59 in 118 appearances, and for Besiktas in the Turkish top flight, with eight in 16.
He also scored twice against Argentina in the World Cup, and United director of football 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.”