The Reds started off on the front foot playing at a high tempo and creating the majority of the chances.
Liverpool did not have to wait long to take the lead. Daniel Sturrige scored on the 20th minute from a questionable penalty decision to take his overall tally to five goals and one assist in his last seven games against United in all competitions.
Firminho doubled the home sides lead scoring in his third consecutive game to give Liverpool an advantage going into the second leg.
Liverpool would have scored again were it not for the heroics of United keeper De Gea who faced eight shots on target and made a string of remarkable saves.
United are now without a win in their last six knockout games in all competitions but will have the chance to redeem themselves in the second-leg at Old Trafford next week.
Liverpool v Manchester Utd
Liverpool confirmed line-up: Mignolet, Clyne, Lovren, Sakho, Moreno, Can, Henderson, Lallana, Coutinho, Firmino, Sturridge
Subs: Ward, Toure, Benteke, Allen, Origi, Smith, Ojo
Man Utd confirmed line-up: De Gea, Varela, Smalling, Blind, Rojo, Schneiderlin, Fellaini, Memphis, Mata, Martial, Rashford
Subs: Romero, Darmian, Riley, Carrick, Schweinsteiger, Weir, Herrera