Harvey Hosein hit an unbeaten 66, but four wickets for Joe Denly helped Kent claim victory by six wickets on the final day at The 3aaa County Ground.
Hosein hit his second half-century of the summer, but spinners Denly (4-36) and Adam Riley (4-68) shared eight wickets in the innings to dismiss the hosts for 270, setting Kent 110 to win.
Zak Crawley led the way for the visitors hitting 34 as Kent claim 23 points from the match, while Derbyshire take five.
Alex Hughes and Tom Lace started the day with Derbyshire on 83-2, still trailing by 78 runs following-on. It only took 20 minutes for Kent to make a breakthrough as Matt Henry got one to move away late to Hughes who edged behind for 16.
Lace, on loan from Middlesex, produced another composed knock on first-class debut. He hit 43, making it 83 runs in the match, off 143 balls with four fours, before he was bowled by Stewart, with Derbyshire 117-4.
Two more wickets fell before the lunch interval as Matt Critchley (1) and Sean Ervine (22) were both dismissed by Riley, who took his innings tally to four. Derbyshire 167-6 at the break with a lead of six runs.
The hosts, however, responded well in the afternoon session as only two wickets fell. Hosein and Hardus Viljoen (23) first shared a 53-run stand for the seventh wicket.
Palladino was the new man and together with Hosein, the pair provided stubborn resistance to the Kent attack adding 72 runs for the eighth wicket and extending the hosts’ advantage to over 100 runs.
But Palladino was unable to see it through to tea when he got a delivery that stayed low from Denly and was lbw for 30, with Derbyshire 264-8 at the interval.
The last two wickets went down not long after the restart, but Hosein remained unbeaten on 66 – his highest score of the season. He faced 147 balls and hit nine fours.
Opener Crawley top scored as Kent chased down the 110 runs they needed for victory. They did lose four wickets, with Hamidullah Qadri (2-34) taking two, but the visitors reached their target with 8.4 overs remaining in the day.
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