Liverpool started well and earned a corner kick inside three minutes and as Agger glanced a header across goal up popped Daniel Sturridge to head home past De Gea and cause Anfield to erupt as he celebrated turning 24 with his third goal in as many games this season while becoming the second player to do so in the Premier League for Liverpool.
The goal proved to be the impetus for an array of chances, Van Persie hit an acrobatic overhead after being played in by Evra and at the other end, Coutinho hit an effort wide from 25 yards. Welbeck attempted to test Mignolet but his shot was an easy take for the Belgian stopper.
United had a great chance minutes later as Young swept in a corner which went across the box and surprised an unmarked Van Persie at the far post who failed to control and finish as the ball bounced to safety.
As the game progressed, United became more aggressive in their attack as frustration began to set in as they went in 1-0 down at the break but early in the second period, they should have levelled when a Liverpool clearance found Young who guided a half-volley towards goal but was superbly denied by Johnson.
Twenty minutes to go and United still pushed to equalise with veteran Ryan Giggs meeting Nani's corner to glance a header towards goal but Mignolet was on hand for an easy catch to protect the lead and he was in action again as Nani danced in from the right and hit a sharp drive which the Belgian palmed away after a superb dive.
The points could have been shared late on as Van Persie had United's best chance of the game as Hernandez slipped a ball through to the Dutchman who failed to establish a good connection and saw his attempt slide into the side netting.
Liverpool could have added to their lead in the final seconds when Sterling aimed for the top corner only to be denied by De Gea but they held on for a famous win, one which puts them two points clear at the top of the table as the only team to have a 100% record while also being the sole team with three clean sheets as the league goes into the international break.