Spy vs. Spy, which debuted in MAD magazine #60 (1961), was the creation of Cuban political cartoonist Antonio Prohías. After drawing one too many unflattering cartoons featuring Fidel Castro, Prohías was forced to flee Cuba for the United States in 1960. He was soon hired by MAD publisher Bill Gaines and worked for the magazine until ill-health forced his retirement in 1987. A series of other cartoonists including Bob Clarke (with gag writer Duck Edwing) and Peter Kuper have kept the feature going to the present day.
Here is the very first Spy vs. Spy that appeared in MAD magazine:
Note that the earliest strips resulted in stalemate. Eventually the spies would take turns being the victor or defeated. Here are a few more such strips from 1961: