Swardeston's fifth foray into the last 16 of one of the prestigious countrywide competitions in six years will see them visit Ealing.
The East Anglian Premier League leaders booked their place in the next stage of the ECB National Championship by winning at Kibworth on Sunday.
Lewis Denmark top-scored with 54 as all of the visitors' top five reached double figures in Leicestershire. His half-century helped Swardeston set a target of 198 for victory.
Michael Eccles struck with the hosts' reply on nought and when he trapped their opponents' other opener for one it looked like a routine triumph was going to follow.
Aadil Ali (63) did his utmost to try and make a fist of it but support from elsewhere was never really in the offing. Eccles claimed another scalp, while Sam Thelwell took two wickets, to help restrict Kibworth to 144-8 off their 45 overs and seal a 53-run victory.
Swardeston's fifth-round trip to Ealing will take place on Sunday, July 12.
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