Sarmiento, 19, was making his first Premier League start for Albion last week at the London Stadium but his debut turned to disaster after just 13 minutes.
The Ecuador international held back tears as he was escorted off the pitch by two physios.
Sarmiento now faces around three months on the sidelines as the former Charlton and Benfica attacker works his way back to full fitness after his surgery.
There had been reports from Ecuador that he had already had surgery but Albion head coach Graham Potter said that was incorrect.
"Jeremy is a more serious one which is a blow for him and the team," said Potter.
"As far as I know he has not had surgery. I'm pretty sure it is planned. The recovery time will be similar to Danny Welbeck, I guess so three or months.
"We are trying to get to the bottom of how and why it has happened because that is a big concern for us. We will do our best to work it out.
"Jeremy and Adam (Lallana) did not play against Leeds so I don't think it is fatigue."
It's been a busy time for the Albion physios as Adam Lallana (thigh), Adam Webster (calf) and Lewis Dunk (knee) were all injured at the London Stadium and are ruled out for Tottenham this Sunday at the Amex.
Leo Trossard is also a major doubt after he was stretchered off in the 1-1 draw at Southampton last Saturday with an elbow injury.
Shane Duffy will miss out this weekend as he's suspended following his fifth booking of the season last Saturday.
Defender Joel Veltman is fit to play despite taking a whack to the face at St Mary's. The Dutch international was subbed in the second half after the injury but Potter said it was for tactical reasons.