Argentina international Rojo, who endured a brief spell on the sidelines earlier this season after dislocating his shoulder in November, was expected to face Brendan Rodgers' men at Old Trafford.
However, he picked up an injury in training before the game and was replaced by Phil Jones - who himself returned from a prolonged absence due to a shin problem.
It is the latest in a series of defensive setbacks for manager Van Gaal this term, with the likes of Chris Smalling, Rafael da Silva, Luke Shaw and Jonny Evans having all spent time on the treatment table.
And Van Gaal conceded his disappointment at his inability to select from a fully fit squad.
"Rojo was good. Only yesterday he was in a training session and he got injured and then I had to change my line-up again," Van Gaal said. "It's what I have to do every week.
"It's more easy when all the players are fit and you can select the best player you think [should play], but we cannot do that at the moment."
However, the Dutchman paid tribute to his squad for sticking to the style of football he is trying to employ at United, after the victory over bitter rivals Liverpool made it six straight Premier League wins.
"The most important thing is our philosophy, and when you stick by your philosophy and you follow the principles of that philosophy, every player can do that, because every player has played already in that philosophy.
"It's amazing that we can do that because it's not so easy."
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