United beat Liverpool 1-0 9th January 2011 in the FA cup Kenny Dalglish's first game as manager prior to that they beat them in the league 3-2 at old trafford.
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As of July 3rd 2012, Manchester City's most recent victory over their Manchester rivals in any competitive match was their 1-0 home league win on April 30th 2012.
Saturday, 20th august. Liverpool won 0-2 thanks to a own goal from a arsenal defender and a Luis Suarez from inside the box
This score has never occurred in a Liverpool v. Manchester United match, although Liverpool beat Manchester United 1 - 4 playing away from home on March 14, 2009.