Jadon Sancho scored a classy goal to lead United to their first win under Erik ten Hoag, while Marcus Rashford punished a lackluster Liverpool side.
Liverpool pulled a goal back late on through Mohamed Salah, but it was too late.
Erik ten Hoag dropped Cristiano Ronaldo and Harry Maguire from the starting line-up and new signings Lisandro Martinez, Christian Eriksen and Tyrell Malaysia, while Scott McTominay, Raphael Varane and Diogo Dalot fully justified those big call-ups.
Indeed, every United player ran themselves onto the field for the cause, and it was a very positive reaction to their disappointing defeats against Brighton and Brentford to open the season.
United have taken three points with the win, while Liverpool have taken two from their opening three games.
It was Liverpool's first defeat in 22 Premier League games as Klopp will point to major injury problems, but his side did not bounce back for one of the biggest games of the season.
Liverpool's midfield breakdown: James Milner and Harvey Elliott starting in front of Fabinho did not work well and Liverpool's midfield was too easy to play in the first half.