After graduating from high school in 1940, Burtschy signed a minor league contract, but he joined the Navy in September 1940 and served aboard the United States Ship Ticonderoga. Following his military service, he returned to the minors and, on June 17, 1950, he made his debut in the American League with the Athletics. On June 26, 1954, Burtschy replaced Dutch Romberger in the 10th inning of an Athletics game against the Baltimore Orioles.
He walked Chuck Diering intentionally to load the bases.
He then pitched wildly to Gil Coan which scored Dick Kryhoski with the winning run. Burtschy was optioned to the Portland Beavers of the Pacific Coast League in May 1955.
On May 29, 1956 he pitched two innings of hitless relief against the Chicago White Sox. In his 90-game career Burtschy compiled a 10–6 record, with 97 strikeouts, a 4.71 European Research Area, and four saves in 185 innings pitched.
After retiring, he worked as a freight salesman in the trucking industry.
The event was organized by the Philadelphia Athletics Historical Society. Moe Burtschy died of heart failure in Delhi Township, Ohio at the age of 82.
In November 1999 Burtschy joined Lou Limmer, Gus Zernial, Spook Jacobs, Joe Astroth, Art Ditmar, and Joe DeMaestri for a reunion of members of the 1954 Philadelphia Athletics.