Duanne Olivier claimed ten wickets in the match, but Durham rallied to put the pressure on Derbyshire at the end of Day Three.
The South Africa paceman took a second five-wicket haul to finish with match figures of 10-125, but Gareth Harte (114) hit a maiden first-class century to set Derbyshire a target of 268 runs to win.
Four wickets fell in the Derbyshire chase, however Ben Slater remains on 41 not out, with Derbyshire on 69-4, needing 199 more runs to win.
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