United enjoyed a dream start as they earned a penalty after just 35 seconds when Daniel Agger fouled Dimitar Berbatov and Ryan Giggs slotted the spot-kick home in resounding fashion to give Kenny Dalglish in the Reds hotseat a baptism of fire.
And after Maxi Rodriguez had tested Tomasz Kuszczak with a swerving shot, Liverpool's plight got far worse on the half-hour mark when Steven Gerrard saw red for a two-footed challenge on Michael Carrick.
Javier Hernandez almost doubled United's lead after the break but he flashed a header just wide, while Berbatov blazed over from 12 yards.
Liverpool refused to lie down, though, and put up a spirited fight, with Ryan Babel's low drive being well saved by Kuszczak, who then made an even better stop to deny a Fabio Aurelio free-kick.
United hit back and Pepe Reina made an outstanding triple save to keep the score at one, but United progress to the next round as they aim for glory in this competition for the first time since 2004.