Sussex recovered from a terrible start to post 215 for seven from their 40 overs with Nash smashing 95 off 106 balls and Joyce hammering 90 off 91 deliveries as they piled on 173 runs for the fourth wicket.
Australia paceman Trent Copeland took career-best figures in one-day cricket of five for 32, but the the home side never looked like keeping up with the run-rate as they collapsed to 122 for nine from 26.3 overs, with Sussex winning by 61 runs on the Duckworth-Lewis method.
Kent enjoyed a successful trip abroad but were made to work harder than they thought to secure a seven-wicket win over the Netherlands.
After bowling their hosts out for 101, victory was always going to be assured for James Tredwell's men, but despite making their target with more than 16 of the rain-reduced 39 overs to spare, their run chase was hardly a swashbuckling one.
The Netherlands had them at nine for one, 39 for two and 53 for three to give them a series of minor scares, but ultimately they had enough to get over the line.
A careful and unbeaten 45 from Brendan Nash got them there, with Ben Harmison alongside him at the end.