Liverpool had trounced Arsenal 5-1 at Anfield in the Premier League last weekend, but the London club took revenge through goals in each half by Alex Oxlade-Chamberlain and Lukas Podolski.
Victory at their Bramall Lane ground took the third tier club into the last eight, where they will now face Sheffield Wednesday if their local rivals get past Charlton in a re-arranged fixture.
Gus Poyet's Black Cats have already reached the Capital One Cup final and they were rarely tested as Craig Gardner's fine strike settled the tie.
The 29-year-old made his comeback after three months out with an ankle injury, playing the last 25 minutes of Wednesday's 5-0 romp at Hamburg in the German Cup quarterfinals.
The 26-year-old was the subject of two close-season bids from Old Trafford but chose to stay at the Nou Camp.
The manager's quip was a reference to Jose Mourinho's depictions of his Chelsea side as a "little horse" and Manchester City as a "jaguar".
It was the London club's first win at St James's Park since 2004 and left them with outside title hopes, occupying fifth place in the Premier League, seven points behind leaders Chelsea.
Johnson's team climbed up to third in the Premiership on Wednesday evening, courtesy of a 2-0 victory over visiting Mpumalanga Black Aces in Atteridgeville.
United were held to a 0-0 draw at Arsenal on Tuesday and their disappointment was compounded by the news that Champions League qualification rivals Liverpool and Tottenham Hotspur had both won.
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