Resuming 102 behind on 195-6, the hosts lost Matt Critchley for 20 to Steve Magoffin (2-44), while Shiv Thakor (81) added 23 to his overnight score.
Tom Taylor fell to Ollie Robinson (4-110), but Tony Palladino (29 not out) and Ben Cotton's (26) last-wicket stand of 51 made Sussex bat again.
Needing 11 to win, the visitors wrapped up victory inside the first over.
Sussex move on to 70 points from six matches, while Derbyshire are on 52 points but still looking for their first win of the campaign.
Sussex beat Derbyshire by 10 wickets to secure their first victory of the season in Division Two.