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.

The 21-year-old became the youngest ever back-to-back champion in the premier class as he clinched his second world title on the Motegi track.
Repsol Honda rider Marquez won the title with an unassailable lead of 312 points in the overall standings with three races to go.
"Just two years in MotoGP with two titles is incredible, " said Marquez.
Spain's Jorge Lorenzo won the race 1.638 seconds ahead of Marquez with Italy's Valentino Rossi third.
Rossi moved up to second place in the championship ahead of Marquez's team-mate Dani Pedrosa with Lorenzo fourth.
Marquez won the the title with 11 victories in 15 races, including a record-equalling 10 in a row.
He is also the youngest rider to win two consecutive world championships, beating the previous record held by Britain's Mike Hailwood in 1963.
Marquez did not make the best start off the line from fourth on the grid.
But he hunted down the front group and overtook Pol Espargaro, Dani Pedrosa and Andrea Iannone in the opening laps.
He then picked off Andrea Dovizioso who started on pole and won a mid-race battle with Rossi, overtaking at the second attempt on lap nine for second place.
Marquez made an attempt to catch Lorenzo, but the Yamaha rider responded and Marquez settled for second.
Pedrosa finished in fourth place just 0.5secs behind Rossi.
"The race doesn't really matter today, " said Marquez, who has also won world titles at Moto2 and Moto3.
"Maybe I wasn't riding like always, as I felt a little different. It was hard to overtake and I didn't want a mistake. I had two mistakes in the last races but we got the title. It's incredible to take it at home. Thanks to my family and Honda."
"Maybe for the people it looks easy because I'm always smiling, but there is a lot of pressure. I'm really happy with my team and family and fans, they're all pushing me. It's like a dream right now."
The next race is Phillip Island, Australia on 19 October.
Motegi MotoGP result:
1.  Jorge Lorenzo (Spain) Yamaha   42:21.259 secs
2. Marc Marquez (Spain) Honda        +1.638
3. Valentino Rossi (Italy) Yamaha       +2.602
4. Dani Pedrosa (Spain) Honda           +3.157
5. Andrea Dovizioso (Italy) Ducati      +14.353
6. Andrea Iannoneita (Italy) Ducati     +16.653
7. Stefan Bradl (Germany) Honda       +19.531
8. Pol Espargaro (Spain) Yamaha        +19.815
9. Bradley Smith (Britain) Yamaha   +23.575
10. Alvaro Bautista (Spain) Honda   +35.687
Andrew Gale and Tim Bresnan hit half-centuries for Yorkshire at Headingley to deepen Sussex's relegation fears.
[NEXT_CONCEPT]
Spaniard Marc Marquez retained his MotoGP championship title with a second-place finish in Japan on Sunday.