United fans began the night with a protest march aimed at the club's owners, the Glazer family, after a disastrous start to the season, but the Red Devils moved above Liverpool in the table with their first league win over Jurgen Klopp's men since 2018.
A series of errors at the backline of Liverpool eventually led to a nice one-two between Christian Eriksen and Elanga before the latter put the ball in the box for Sancho.
He sat veteran defender James Milner down the box with Alisson Becker diving the wrong way before calmly slotting it into the bottom corner to give Man Utd a 1-0 lead.
It was the much-needed relief for the Old Trafford faithful after a humiliating loss to Brentford and Brighton in the first two games.
Liverpool who were missing several players on injury and new signing Darwin Nunez due to suspension were less dangerous going forward and crosses from Andy Robertson and Alexandre Arnold were wasted in the final third.
Manchester United got a comfortable lead in the second half as Anthony Martial teed up Rashford, he raced down to the Reds post and buried it past outrushing Alisson.
It was much of a game-set-match after doubling the lead until Mohamed Salah got on the end of a parried ball by De Gea and skilful heads it into the net.
It was more of a consolation for Jurgen Klopp's side as time-wasting tactics were used by United to run the game down.
Now Liverpool have gone three games without a win while Man Utd ends their streak of loss with a narrow win over the Reds.
Man Utd will next travel to Southampton while Liverpool will play newbie Bournemouth at Old Trafford.
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