He had been in promising form for the Clarets in recent weeks, with the lay-off coming as a blow to the side ahead of a busy Christmas period.
An early goal from Demarai Gray, set up by Riyad Mahrez, earned Claude Puel all three points.
Burnley manager Sean Dyche explained after the match that the injury didn't look good.
Ireland global Brady, who has scored once and provided two assists in 15 league appearances for Burnley this campaign, was carried off the pitch on a stretcher during last weekend's 1-0 defeat at Leicester City.
On Wednesday the news Burnley fans have been fearing has come in.
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Robbie Brady faces a substantial period on the sidelines after sustaining a knee injury.
Since then, the 25-year-old Brady has spent the last few days in hospital undergoing scans and medical experts are believed to have diagnosed a tear of his patellar tendon.
Burnley FC have released an official statement after Republic of Ireland midfielder Robbie Brady suffered a nasty knee injury in Burnley's weekend defeat to Leicester City..
"He will be given all the time and care he needs on his recovery and supported throughout from all of us." .
Dyche is expected to give more details on Brady's situation at some stage today.