Resuming on 84-1, Varun Chopra and Ian Bell, who hit the first six of Warwickshire's season, moved the total on to 105 before the Essex born batsman was caught by Rudolph from the bowling of Tim Bresnan.
Bell passed the fifty mark but on 54, he became Ajmal Shahzad's second victim when he was caught by Lyth.
Jon Trott had played some excellent shots and hit 15 fours in a knock of 88 but he gave McGrath a catch from a ball by Hannon-Dalby to become the fourth Warwicks player out with the score on 231.
Tim Ambrose came and face four balls before being trapped by Hannon-Dalby without scoring.
Hannon-Dalby took his third wicket of the innings when he bowled Chris Woakes for two and picked up his fourth as Neil Carter was caught by Gale for nine.
That left Warwicks on 261-7 and Jim Troughton continued to keep adding runs. The Stratford man had made 78 when he was bowled by Patterson shortly into the final session of the day.
Jim Troughton
Imran Tahir could only add three runs before he was caught by Bairstow to give Steven Patterson his second wicket to take the score to 329-9.
Naqaash Tahir and Andrew Miller were at the crease. Tahir had made 34 before he was out lbw to give Hannon-Dalby figures of 5-68 and meant that Warwicks were 347 all out.
Yorkshire required 291 runs to win with an hour and a whole day to get them in.
Adam Lyth and Joe Sayers opened the batting and had taken the score to 48 before Sayers was out, caught by Ambrose off the bowling of Naqaash Tahir. Yorks closed on 57-1 needing another 234 for victory.
You can hear ball by ball commentary by clicking on the link on the BBC Coventry & Warwickshire website.
Picture copyright of Action Images. Credit Action Images/ Carl Recine