Starting the fourth day with a lead of just 11 with only three second-innings remaining, hosts Derbyshire could only add 30 more runs before they were bowled out for 219, Steve Magoffin (four for 44) claiming two of the wickets to fall.
Sussex then lost Chris Nash to a first-ball duck as they chased 42 for victory but Luke Wells (23 not out) and Michael Yardy (18no) quickly saw them past the winning line in less than six overs as Derbyshire slumped to a fourth successive defeat.
Nottinghamshire are also closing in on victory against Surrey at Trent Bridge.
Needing 259 to win, Surrey lost five wickets as they went to lunch on 138 for eight.
Paul Franks dismissed both overnight batsmen Rory Burns (53) and Zander de Bruyn (14) to start the collapse before Harry Gurney weighed in with the wickets of Gary Wilson (nought), Gareth Batty (four) and Steven Davies (28).
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- Steve Magoffin