Beginning the day 52 for one, with a lead of 186, the hosts lost three batsmen for the addition of only 34.
Rashid's knock, coming off 69 balls, guided his side to 158 for five at lunch, an advantage of 238, while Surrey's Gary Keedy took his tally to four for 70.
Derbyshire crashed to 94 all out to leave themselves teetering on the brink of defeat at home to Somerset in the bottom-of-the-table clash.
Resuming on their overnight 36 for two, they collapsed as Steve Kirby added two wickets to the couple he had already taken to finish with four for 18 off seven overs. Craig Meschede provided support with three for 17 as Derbyshire lost their last eight wickets for 56 runs.
That set their opponents a victory target of 145, and Nick Compton (15 not out) helped reduce that by 30 at the interval.
Sussex finally brought their first innings to an end once Ed Joyce had completed his double century at home to Nottinghamshire.
The former England batsman resumed day three on 129 not out, with his side 363 for four, and he progressed to an unbeaten 204 in taking Sussex to 506 for five declared.
Rory Hamilton-Brown's 113 was overshadowed somewhat as a result before Nottinghamshire reached 20 without loss, Riki Wessels scoring 15no.
In Division Two, Essex are hanging on as they seek to avoid defeat against Leicestershire after slipping to 155 for eight, a lead of only 101 with just two sessions of the match to go.