Tony Palladino claimed his 300th wicket for Derbyshire, but Stuart Poynter (170) and Michael Richardson (115) put Durham in control at The 3aaa County Ground.
Their 278-run stand helped Durham recover from 175-5 to 453-6 and into the lead, ultimately all out for 520 with a lead of 93.
Ben Slater and Luis Reece (one not out) survived the final two overs of the day to take Derbyshire to 1-0, trailing by 92 runs.
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