Joyce made 92 and Brown 93 as Sussex were dismissed for 356 having resumed on 104-3, a lead of 260.
Steve Magoffin completed another excellent day for Sussex as he removed Alex Lees before rain and bad light brought a premature end with Yorkshire on 27-1, still 233 behind.
Joyce put on 51 with Rory Hamilton-Brown for the fourth wicket before Hamilton-Brown was trapped lbw by Liam Plunkett for 26.
Joe Gatting made a quickfire 20 before Brown joined Joyce to put on 120 for the sixth wicket.
Joyce - who had dropped down to No.4 for the first time for Sussex - finally fell seven short of his century after hitting 16 fours in his 140-ball innings.
The Sussex skipper was caught by Adam Lyth off the bowling of Jack Brooks who went on to remove Chris Jordan for a golden duck next ball.
Debutant Brooks also picked up the wickets of Magoffin and Brown - who made his 93 off just 95 balls, including 14 fours - to finish with figures of 4-76 while Ryan Sidebottom wrapped up the Sussex innings by trapping Monty Panesar lbw to post figures of 4-72.
Magoffin struck in the fourth over of Yorkshire's second innings as he had Lees caught at slip by Jordan but Phil Jacques and Lyth survived the next four overs before play was suspended.