Scarbrough left the field during the second half against Gloucester on Sunday and scans revealed that he had suffered a ruptured medial cruciate ligament in his knee.
He underwent surgery late on Monday night and is not expected to return until the end of January.
"It's a significant injury from our point of view," said director of rugby Alan Gaffney.
"You have to feel very sorry for 'Scabs' who had started the season very well and looked in good form. As soon as the scans revealed the problem he had surgery straight away but it will be the end of January before we can expect to see him back on the training pitch.
"It's a blow but something we will have to deal with. We managed to escape pretty much unscathed in terms of long-term injuries last season but it looks like we may have used up all our luck!"
Kevin Sorrell picked up a calf injury during the same game which will rule him out for up to three weeks.