Joe Gatting was stranded on 99, Adam Lyth having 109 not out for the Tykes.
But it was the 29-run stand between Joyce (21 from 34) and Joe Gatting (25 from 22) which carried Sussex past the 100-run mark.
For Gatting, 21, the nephew of former England skipper Mike and son of ex-Arsenal and Brighton footballer Steve, it was comfortably his highest score in one-day cricket.
Off-spinning all-rounder Nabi struck a rapid century and paceman Hassan took 3-9 off six overs to earn rave reviews from former England captain Mike Gatting, who also took part in the match.
And, when the rain set in, despite striking 13 fours and three sixes from his 90 balls, that left a frustrated Gatting just one short of that first one-day ton.
Yardy and Joe Gatting, on his Championship debut, got through 5.4 overs before the rain came down to end the day, leaving Sussex to resume on day three 225 behind.
Gatting, having started confidently, moved to his 50 before hitting spinner David Wainwright for a midwicket six as he put on 58 for the fourth wicket with his skipper Mike Yardy.
Gatting, whose 1986-87 tourists were written off in the media before winning Down Under, was at Lord's to see the delicate Ashes urn begin its three-month tour of six Australian cities.
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Former England pace bowler Saj Mahmood ripped out Championship debutant Joe Gatting's off and middle stumps, but Yardy looked untroubled, as he moved to within 114 of his 1, 000 four-day runs for the season.
That was not the view of Mike Gatting, the manager of the England A tour to Bangladesh late last year, who reasoned that the standard of cricket in the country is not sufficiently high to compete at Test level.
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