After choosing to bat, captain Michael Thornely and wicketkeeper Brown provided the main impetus of Sussex's innings with a third-wicket stand of 83 runs, after Matt Machan and Joe Gatting had fallen early on.
Thornely was caught behind for 56 and Brown's innings of 58 came to an end when he missed a sweep off Faisal Hossain.
Bangladesh's new captain Mashrafe Mortaza was the most effective bowler, taking 2-32 off his nine overs, with three team-mates also picking up two wickets each.
But after a steady start to their reply, Bangladesh soon lost their way as they slid from 36-1 to 49-5.
Tamim Iqbal, who scored successive hundreds in the Test series against England, was the first man out, caught behind off the bowling of 19-year-old James Thorpe.
Jahurual Islam was next to go, again caught behind by Brown, off the first ball bowled by 20-year-old Will Adkins.
Once James Anyon claimed the wickets of Mushfiqur Rahim and Faisal Hossain within three balls, there was no way back for the Bangladesh.

