Michael Stoneman hit his fifth first-class century before Durham set Yorkshire a victory target of 336 at Chester-le-Street.
Durham declared at 275-4 in their second innings and Yorkshire openers Adam Lyth (9no) and Joe Root (6no) survived the final seven overs of day three to leave the visitors on 17-0 at stumps, needing another 319 on the final day.
Stoneman plundered 11 fours and a six in his 109 and received good support from Keaton Jennings during an opening stand of 123 in 40.2 overs at the start of day three.
Jennings was caught behind off Yorkshire seamer Tim Bresnan (2-70) for 48 and Will Smith (4) went to the same bowler later in the same over.
Stoneman brought up his three figures during a 68-run, third-wicket stand with Dale Benkenstein.
Liam Plunkett (2-49) finally ended Stoneman's resistance and he followed up with the wicket of Ben Stokes (7) to leave Durham on 207-4.
Benkenstein (61no) and skipper Paul Collingwood (36no) powered the north-east county up to their declaration without further loss.