Finch was down two to Martin Kaymer after eight holes of the final round-robin match but pulled even with birdies at Nos. 9 and 11. Kaymer went back up again on 12 but had a bogey in the 14th hole that allowed Finch to get square.
Finch managed a par on the par-5 18th hole, where Kaymer again had a bogey. The 1-up win put Finch into Saturday's Round of 16.
Defending champion Ian Poulter won his Friday match 4-and-3 over Tom Lewis and took the Player Group. Lewis earlier in the day beat John Senden 1-up and he claimed the group's other knockout round spot.
Rafael Cabrera-Bello, with wins over Finch and Kaymer, won the Ballesteros Group.
Retief Goosen took the Palmer Group with a 2-0 mark while Nicolas Colsaerts finished second by taking a tiebreaker against Charl Schwartzel.
Saturday's schedule opens with Garcia playing Lewis followed by McDowell going against Finch. Lawrie and Bjorn have the third match of the day with Goosen and Rock off fourth.
Snedeker plays Villegas and Rose meets Colsaerts while Poulter faces Quiros and Cabrera-Bello is paired against Karlsson in the other Round of 16 matches.
Winners advance to the quarterfinals Saturday afternoon with the semifinals and finals set for Sunday.