The transfer includes a guaranteed £66.3m payment to the German club, with the remainder in add-ons.
The 22-year-old has scored 39 goals in 87 appearances for the Bundesliga side, and heads to Old Trafford with a growing reputation as one of Europe's top forwards.
The 6ft 5in Slovenian scored 13 times and assisted five goals in 33 league appearances last term, and has been clocked at a top speed of 35.7km/h - one of the fastest in Germany - as well as boasting the highest aerial duel success rate among Bundesliga strikers at 57.4 per cent.
"The history of Manchester United is obviously very special, but what really excites me is the future," said Sesko in a press release.
"When we discussed the project, it was clear that everything is in place for this team to continue to grow and compete for the biggest trophies again soon.
"From the moment that I arrived, I could feel the positive energy and family environment that the club has created. It is clearly the perfect place to reach my maximum level and fulfil all of my ambitions.
"I cannot wait to start learning from Ruben and connecting with my teammates to achieve the success that we all know we are capable of together."
The announcement will sting for Newcastle fans, who had previously seen their side miss out on signing Hugo Ekitike, and are likely to see Alexander Isak depart this summer.
Man Utd, on the other hand, are looking forward to welcoming Sesko alongside fellow newcomers Matheus Cunha and Bryan Mbeumo, bringing their total spending on attack to over £200m this summer.
Only four other teams scored fewer than United's 44 goals in 38 Premier League games last season as the English giants endured their worst campaign for 51 years, finishing 15th in the table.
Between them, Sesko, Cunha and Mbeumo combined for 58 goals last season.
"Benjamin possesses a rare combination of electrifying pace and the ability to physically dominate defenders, making him one of the most exceptional young talents in world football," United's director of football Jason Wilcox added.
"We have followed Benjamin's career closely; all of our data analysis and research concluded that he has the required qualities and personality to thrive at Manchester United.
"Working under the guidance of Ruben and our excellent performance team, Benjamin is joining the perfect environment to support him to reach his world-class potential."
Sesko is likely to make his debut for the Red Devils in their Premier League opener against Arsenal next weekend.