Finch (114no off 64 balls) muscled seven sixes – including four in a row off David Wiese (1-60) in the 30-run 18th over – and as many fours to propel the hosts to 193-2 in the capital.
Jason Roy (50 off 33) cracked four maximums of his own in an opening stand of 102 with Finch, including one off the unfortunate Wiese that flew out of the ground and onto the roof.
Sussex could only muster 176-7 in reply with Surrey skipper Gareth Batty claiming figures of 4-24, including Chris Nash (53) after the Sharks opener had notched his fifth fifty of the Blast campaign.
Surrey’s triumph lifts them from bottom in the South Group to sixth,
– Jamie Porter (4-20) restricting the hosts to 121-8 before Paul Walter (18no) and Simon Harmer (11no) saw Eagles home in the 17th over after they had slipped to 96-7.
at Taunton, despite Johann Myburgh’s 87, as they finished on 182-6 chasing the Welsh county’s 183-6, in which Nick Selman hit 66 off 42.
Marchant de Lange was the hero for Glamorgan, conceding just six of the eight runs Somerset required in the final over – Peter Trego’s last-ball boundary in vain – as the visitors earned a last-four spot.
North Group leaders Nottinghamshire also secured a quarter-final berth – at New Road.
Christian (22) and Billy Root (37) put on 50 for the seventh wicket after Notts had stumbled to 80-6, before the captain bagged three wickets to limit Rapids to 134-8, Harry Gurney (2-18) and Jake Ball (2-22) also impressing.
– the Bears coasting past Jets’ 145-8 in with 5.1 overs remaining thanks to half-centuries from openers Ed Pollock (52 off 25) and Dom Sibley (51no off 38).