Bairstow struck 161 not out as Yorkshire declared on 470-7, before Sussex safely negotiated ten overs to finish the day on 21-0 - still 133 runs behind.
Yorkshire had started the day on 208-3 and Jack Leaning was the first to fall, a run short of his maiden century, when he was caught and bowled by James Tredwell with the score on 264, after hitting 12 fours from 257 balls.
Big-hitting Australian Aaron Finch was out lbw to Steve Magoffin for three to make the score 267-5.
However, Tim Bresnan then joined Bairstow at the crease and the duo put on 138 for the sixth wicket.
Bresnan was bowled by Magoffin - his fourth wicket of the innings - for 68, after striking nine fours and one six off 169 deliveries.
Azeem Rafiq was then caught by Jon Lewis off Tredwell, which made the score 428-7.
Bairstow, who struck eight fours and three sixes from 290 balls, then added a further 42 with Richard Pyrah (20 not out), before Yorkshire declared.
Magoffin finished with 4-81 from 42 overs and Tredwell claimed 3-158 off 59 overs.