Sussex could have taken third place had they won at Hampshire, but rain forced their match to be abandoned.
Somerset's fixture at Middlesex was also called off, meaning Somerset finish second in the South Group.
Essex finished top of the group after beating Kent by 57 runs on the Duckworth-Lewis method at Canterbury.
They are joined by North Group winners Worcestershire in going straight through to the semi-finals.
Somerset will host Nottinghamshire in the quarter-final on 13 June with the winner to face Essex at Chelmsford three days later.
Yorkshire play Surrey at Headingley in the other quarter-final, also on 13 June. The winner travels to Worcestershire on 17 June.
At Canterbury on Wednesday, Varun Chopra scored 83 from 98 deliveries, while England Test opener Alastair Cook made 54 as the visitors reached 307-6 from their 50 overs.
Kent lost three early wickets before rain caused play to be stopped after 11 overs had been bowled, with the hosts on 50-3 - well short of their adjusted target.
Surrey captain Gareth Batty told BBC Radio London:
"It was 100% the right decision [to call the game off], the water's coming over your shoes out there.
"There's nothing you can do, you just move on to the next game."
Surrey qualified for the One-Day Cup knockout stages despite rain causing their game at Gloucestershire to be called off without a ball being bowled.