Shaw, 25, limped off during the closing stages of the match and was replaced by Axel Tuanzebe.
The England defender is expected to miss the upcoming Premier League games against West Bromwich Albion, Southampton and West Ham United as well as Champions League clashes with Istanbul Basaksehir, Paris St Germain and RB Leipzig.
Shaw had played every minute of United's Premier League campaign before the injury and his absence will pave the way for new recruit Alex Telles, who tested positive for COVID-19 last month but has since recovered, to make his Premier League bow.
