In 1903 with the Boston Americans (later the Boston Red Sox) winning the World Series.
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There's some debate, but the answer is that it was founded in either:
1871, if you believe the National Association counts as a Major League (which is debatable). Teams folded and moved midseason, teams played widely varying numbers of games in a season (for example, in 1875, the Boston Red Stocking played 79 games, while the Philadelphia Centennials played 14), hot dog vendors were called in to pitch in blowouts...it was quite ragtag in many ways.
1876, if you believe it began with the conception of the modern National LEAGUE (not to be confused with its unrelated, similarly-named predecessor), or
1901, if you believe that Major League Baseball only truly existed once both American and National Leagues were around as we know them.
Most scholars believe that it is 1876, but arguments can be made in any of the three cases. One might even argue that it was not truly a Major League until the color barrier was broken in 1947 - how could it be considered the best of the best, if some of the best athletes were not allowed to participate?
The first World Series was played in 1903. It featured the Boston Americans versus the Pittsburgh Pirates. Boston won the series 5-3.