Brown was taken off on a stretcher in the early stages of the game in Austria with a problem that has yet to be diagnosed.
Things quickly went from bad to worse for the Scottish champions, as Steffen Hofmann's free-kick skimmed the top of Virgil van Dijk’s head and beat goalkeeper Lukas Zaluska with just seven minutes on the clock.
Celtic responded impressively thereafter, though, and Anthony Stokes and Kris Commons went close to equalising in the first half.
Rapid pushed to extend their lead, but Celtic came back into the game and eventually got their rewards in the 70th minute.
Leigh Griffiths' strong run down the right ended with Teemu Pukki scoring at the second attempt.
Celtic now move on to a friendly against LSK Linz on Tuesday, but new coach Ronny Deila will be most concerned by the injury sustained by Brown.