The most runs scored by a single team in the first inning is 16 by the Boston Beaneaters against the Baltimore Orioles in 1894. The most runs scored by both teams in the first inning is 16 held by Boston and Baltimore as described above (Boston 16, Baltimore 0) and the Oakland Athletics and California Angels in a game in 1996 where Oakland scored 13 runs and California scored 3 runs.
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16 by the Boston Beaneaters in a game against the Baltimore Orioles in 1894. Boston won the game 24-7.
The modern day record is 14 by the Boston Red Sox in a game against the Florida Marlins in 2003 (Boston won the game 25-8) and the Cleveland Indians in a game against the Philadelphia Athletics in 1950 (Cleveland won the game 21-2).
The highest number of runs scored in one inning was by the Kansas City Royals in 1979. In the fourth inning on June 14, 1979 the Royals scored 4 off of a grand slam by George Brett.