The No. 9-ranked German needed to make a 3-foot putt to halve the match and force a playoff against the No. 218-ranked Finch, but it horseshoed out.
Kaymer, who won the 2010 U.S. PGA Championship, started the year at No. 4 but hasn't won an event this year. He joins Darren Clarke and Charl Schwartzel as major winners who failed to make the weekend.
Finch made par on the par-5 last and Kaymer should have matched it, but turned away in disbelief after his effort wrapped around the edge of the cup.
Finch's reward is a match against McDowell, who beat Jbe Kruger 4 and 3 and is attempting to recreate his match-play heroics from the 2010 Ryder Cup at Celtic Manor, where he holed the winning putt.
''It was a bit of a dead rubber – the real fun starts tomorrow against Richard,'' McDowell said.
American Brandt Snedeker was reunited with his own clubs Friday for his head-to-head with Branden Grace. He won his first match with borrowed clubs, a putter from the pro shop and a player's backup driver after his clubs were lost because of an emergency landing Monday during his flight to Spain after a passenger had a heart attack.
Bjorn beat Grace on the next playoff hole and will take on Paul Lawrie next.