“No game was ever won by resigning” (or, “You can’t win by resigning”) is a chess adage that is usually credited to the chess grandmaster and journalist Savielly Tartakower (1887-1956). “No one has yet managed to win a game by resigning” has been cited in print since at least 1948 and “You cannot win by resigning” has been cited in 1958.
 
A similar saying is “A winner never quits and a quitter never wins.”
 
     
Wikipedia: Savielly Tartakower
Ksawery Tartakower (also known as Saviely or Savielly Tartakower in English, less often Xavier Tartacover or Xavier Tartakover; 1887–1956) was a leading Polish and French chess Grandmaster. He was also a leading chess journalist of the 1920s and 30s. Tartakower is best remembered for his sharp wit and countless aphorisms which kept his audience bent over with laughter. His books remain very popular even today at all levels.
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Quotations
“No game was ever won by resigning.”
       
Google Books
Nimzovich, the hypermodern
By Fred Reinfeld
Philadelphia, PA: David McKay Co.
1948
Pg. 10:
Tartakover’s ironic aphorism “No one has yet managed to win a game by resigning” sums up the matter.
 
Google Books
Chess Review: the picture chess magazine
Volume 26  
1958
Pg. ?:
White is well versed in the elementary principle: “You cannot win by resigning.”
 
2 February 1964, New York (NY) Times, “Chess” by Al Horowitz, pg. X23:   
But Wade believes in the old adage: “You can t win by resigning.”
 
14 October 1965, New York (NY) Times, “Chess” by Al Horowitz, pg. 44: 
This all proved once again Tartakover’s aphorism “You cannot win by resigning.”
 
22 May 1978, Oregonian (Portland, OR), ‘Evans on Chess,” pg. B4, col. 4:
NEVER SAY DIE
You can’t win be resigning. If your opponent can go wrong, give him a chance to find it.
 
Google News Archive
12 April 1984, Evening Times (Scotland), “Chess” by Gerald Bonner, pg. 28, col. 1:
THERE is an old adage in chess that nobody ever won by resigning.