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 horse-shoed out.
Kaymer, who won the 2010 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 it to 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 Graeme 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.