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The world number nine Kaymer missed a three-foot par putt at the final hole on the Finca Cortesin to hand a place in the last 16 of the 2,750,000 euro ($3.5m) event to the 218th ranked Englishman.
Finch, who is now guaranteed at least 70,000 euros ($89,000) even if he loses at the start of Saturday's knockout stage, only qualified for the event in finishing third in the recent China Open.
As both the first and second-place finisher in China had already qualified, the former Irish Open winner Finch was handed a place in the Spanish field.
Finch will now play former US Open champion Graeme McDowell after he accounted for South African Jbe Kruger 4 & 3.
Defending champion Ian Poulter, 36 cruised into the last 16 with a similar 4 & 3 victory against fellow Englishman Tom Lewis.
However, Poulter had qualified already qualified for the weekend after Lewis had beaten Australia's John Senden earlier in the day.
It meant Senden, with two loses, was out of the event leaving Poulter and also Lewis to go through no matter the outcome of their afternoon encounter.
"In terms of adrenalin and being 100% switched on, I don't think either of us were 100% switched on," said Poulter.
"So it was hard to get that adrenalin rush you normally get in match-play knowing that both of us would go through.
"It was still a classic match-play encounter as Tom found the water at the fourth and then after hitting from the drop zone he flies his 120-yard shot straight into the cup without bouncing.
"But with Tom having to play 36 holes today, he was probably getting a little tired and wanting to get in."
Poulter will now face Alvaro Quiros with the Spaniard easily accounting for Japan's Tetsuji Hiratsuka 4 & 2.
And Poulter remains on target to join some of the greats of the game including the late Seve Ballesteros in successfully defending the Volvo World Match-Play Championship.
Ballesteros won the event back-to-back in 1981 and 1982 as well as 1984 and 1985. Other players to have successfully held onto the prestigious title include Gary Player (1965 & 1964), Hale Irwin (1974 and 1975) and Ernie Els, who captured the event three years running on two separate occasions (1994-96 & 2002-04).
In other matches England's Justin Rose also had no trouble in defeating British Open Champion Darren Clarke 6 & 4.
Rose got off to a flying start to win the first two holes as he seeks to qualify automatically for the European team for the Ryder Cup later this year.
"In the general terms of my career I do have a certain determination to get back into the Ryder Cup team," he said.
Rose will now tackle Nicolas Colsaerts after the Belgian defeated 2011 Masters champion Charl Schwartzel in a play-off.
American Brandt Snedeker defeated South African Branden Grace and will play Colombia's Camilo Villegas.
In other matches on Saturday morning Spain's Sergio Garcia plays Lewis, Sweden's Robert Karlsson plays Spain's Rafael Cabrera-Bello and South African Retief Goosen, who won both of his matches, meets England's Robert Rock.
Results:
Rafael Cabrera-Bello(Spain) beat Richard Finch (Britain) 4&2
Richard Finch (Britain) beat Martin Kaymer (Germany) by 1 hole
Robert Rock (Britain) beat Darren Clarke (Britain) by 2 holes
Justin Rose (Britain) beat Darren Clarke (Britain) 6&4
Retief Goosen (South Africa) beat Nicolas Colsaerts (Belgium) by 1 hole
Retief Goosen (South Africa) beat Charl Schwartzel (South Africa) 4&3
Robert Karlsson (Sweden) halved with Jbe Kruger (South Africa)
Graeme McDowell (Britain) beat Jbe Kruger (South Africa) 4&3
Alvaro Quiros (Spain) beat Tetsuji Hiratsuka (Japan) 5&4
Sergio Garcia (Spain) beat Tetsuji Hiratsuka (Japan) by 4&2
Paul Lawrie (Britain) halved with Camilo Villegas (Colombia) Camilo Villegas (Colombia) beat Peter Hanson (Sweden) 6&4
Thomas Bjorn (Denmark) beat Branden Grace (South Africa) 2&1
Branden Grace (South Africa) beat Brandt Snedeker (U.S.) 4&3
Tom Lewis (Britain) beat John Senden (Australia) by 1 hole Ian Poulter (Britain) beat Tom Lewis (Britain) 4&3
