Edinson Cavani struck with two minutes left on the clock, before Anthony Martial scored on the break in injury time as the home side piled men forward in search of an equaliser.
It was, on the balance of play, a deserved victory for the Reds, who dominated possession throughout and could have been two or three goals to the good in the early stages, after a wave of pressure in the opening 25 minutes or so.
Dean Henderson and his defenders - Axel Tuanzebe, Eric Bailly, Harry Maguire and Alex Telles - kept the club's third consecutive clean sheet in this season's competition, while Nemanja Matic and Paul Pogba had the better of the midfield battle.