The Welsh side had posted 183 for six on the back of a career-best 66 from Nick Selman but Johann Myburgh took the hosts close to the finishing line with a well-paced 87.
England#8217;s Jason Roy also chipped in 50 as he and Finch put on 102 for the first wicket at the Kia Oval.
South Group, Taunton: Glamorgan beat Somerset by one run
Openers Ed Pollock and Dom Sibley did the bulk of the heavy lifting, scoring twin half-centuries as Ryan Pringle took two consolation wickets.
Surrey received a shot in the arm as Aaron Finch#8217;s unbeaten 114, from just 64 balls, carried them past Sussex and in to 9/1 from 12s.