The midfielder pulled up inside the first minute after an innocuous challenge from an opponent as he controlled a bouncing ball.
Brown held the back of his upper leg and had his right knee strapped into the stretcher as he was taken off.
Celtic later tweeted: "The captain is still being assessed by the club's medical staff and the extent of the injury is not yet known."
The injury will be of major concern to new manager Ronny Deila ahead of Celtic's Champions League qualification campaign, which begins against KR Reykjavik in Iceland on July 15.
Celtic fought back for a draw after Virgil van Dijk headed into his own net in the seventh minute following a free-kick.
Substitute Teemu Pukki levelled with 20 minutes left after converting a Leigh Griffiths cross at the second attempt.
The match was the last ever at the Gerhard Hanappi Stadium, which is being knocked down to make way for a new stadium for Rapid.