City reached the last four for the first time in 30 years after scoring the only goal of the match when England defender Micah Richards powerfully headed home a corner after 74 minutes.
Stoke City ended an even longer wait to make the semis for the first time since 1972 when they beat West Ham United 2-1 at the Britannia stadium.
They will face Bolton Wanderers, who beat Birmingham City 3-2 on Saturday when United made sure of their place with a 2-0 win over Arsenal.
Macnhester City coach Mancini, under pressure to end City’s 35-year wait for a trophy, said his team looked sluggish after travelling to Ukraine for a Europa League game on Thursday which they lost 2-0 to Dynamo Kiev.
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