The teenager was hurt by a sliding tackle from Kyle Walker-Peters inside the area, after coming on as a substitute in the second half of the stalemate, with the Reds down to 10 men following Casemiro's red card.
Walker-Peters has already wished Garnacho a speedy recovery on social media.
Posting on Instagram, the 18-year-old also confirmed he has had to withdraw from the Argentina squad, following a senior call-up.
"It is difficult to put into words how I'm feeling right now," he said.
"Unfortunately, I will not be able to help my team and team-mates in the upcoming games, in what is a very important part of the season for us at Manchester United.