Burke was in five successive American Ryder Cup teams from 1951 to 1959. He was the playing captain in 1957 and the non-playing captain in 1973. He had a successful playing record, winning his first 7 matches; two matches in 1951, 1953 and 1955, and winning in the foursomes in 1957, before losing to Peter Mills in the singles. Mills won 5 holes in a row from the 6th to the 10th to be 5 up, finished the morning round 5 ahead and eventually won the match 5&3. Great Britain won 6 of the 8 singles and halved another to win the Ryder Cup for the first time since 1933. Burke was in the 1959 Ryder Cup team but had a hand injury and was not selected for any matches.