Having resumed on 55-1, the hosts lost Gary Ballance (45) and Andrew Lyth (39) early on, before Andrew Gale (67) and Jonny Bairstow steadied the innings.
Bairstow was trapped lbw by Chris Jordan after the break, while Chris Liddle removed Jack Leaning cheaply.
But Gale and Bresnan (50 not out) helped their side reach 298-9 at stumps, a lead of 301.
Sussex would need at least a draw to secure their Division One status for next season if rivals Hampshire can complete victory at Nottinghamshire on Friday.
The visitors started the third day strongly with the ball as Ballance and Lyth both played onto their stumps in the morning session.
Gale's patient innings put Yorkshire back on the front foot, and also saw the 31-year-old surpass 1,000 runs in the County Championship this season.
However the captain's 131-ball knock was finally ended when he was caught by Jordan off the bowling of Liddle, but Bresnan's half-century left Sussex in danger of playing Division Two cricket for the first time since 2010 next season.
Andrew Gale and Tim Bresnan hit half-centuries for Yorkshire at Headingley to deepen Sussex's relegation fears.