Daniel Sturridge gave Liverpool the lead from the penalty spot in the first half, before Roberto Firmino made it 2-0 in the 73rd minute on Merseyside.
The result means that the Reds will take the lead to Old Trafford for next Thursday night’s return leg.
Former Liverpool defender Carragher singled out Firmino, Philippe Coutinho and Adam Lallana for special praise after the game.
However, the former England international still feels that Manchester United are not out of the tie.
Carragher tweeted on Thursday night:
The outstanding Firmino the pick of the bunch with Coutinho & Lallana not far behind! A mention for Clyne always 7/10. Far from over.
— Jamie Carragher (@Carra23) March 11, 2016
Liverpool now have a free weekend to prepare for next week’s return leg.
Manchester United, meanwhile, take on West Ham United at Old Trafford in the FA Cup quarter-finals on Sunday.
MORE: The latest football news