O'Donoghue was primarily a starting pitcher during the first half of his major league career, and almost exclusively a reliever during the second half. From 1963 to 1967 he started in 93 of his 139 games, and from 1968 to 1971 relieved in 115 of his 118 games.
He struggled greatly during his first four minor league seasons (1959-1962), from rookie ball to Double-A. He had a combined record of 26-39 with an earned run average of 5.54. In 499 innings pitched he had given up 307 earned runs, struck out 360, and walked 358. In 1963, at the age of 24, it all began to come together. Pitching in the Eastern League and Pacific Coast League, he had a combined record of 14-11 with an ERA of 3.10, leading to his call-up to the pitching-starved Athletics.
O'Donoghue made his major league debut on September 29, 1963...the last day of the season. He was the starting pitcher in a home game against the Cleveland Indians at Municipal Stadium. He gave up just two runs (one unearned) in six innings, but was the losing pitcher as Jim "Mudcat" Grant and the Tribe prevailed, 2-1.
His first major league win came on May 12, 1964 at Dodger Stadium. He started and pitched the first seven innings against the Los Angeles Angels, giving up two unearned runs, and the A's won by a score of 6-2. John Wyatt saved the game for him with two scoreless innings.
O'Donoghue's finest major league effort was against the Detroit Tigers on August 19, 1967. He pitched a one-hit complete game shutout at Tiger Stadium that day, striking out 11 and walking only 2. The Tigers had such players as Dick McAuliffe, Al Kaline, Willie Horton, Bill Freehan, Eddie Mathews, and Norm Cash in the lineup, but O'Donoghue was almost untouchable. Freehan got the only Tiger hit, a 2nd-inning single, as the Indians won 5-0.
Even though O'Donoghue was named to the American League All-Star team in 1965, it probably does not qualify as his best season. He was 9-18 with a 3.95 ERA, and tied for the league lead in losses with Dave Morehead and Bill Monbouquette of the Boston Red Sox. In 1967 he was 8-9, 3.24 and had his lowest career WHIP (1.171). Then, in 1969, he relieved in 55 games for the Seattle Pilots and was 2-2 with 6 saves and a 2.96 ERA in 70 innings.
Career totals for 263 games played (257 as a pitcher) include a 39-55 record, 96 games started, 13 complete games, 4 shutouts, 56 games finished, and 10 saves. He allowed 340 earned runs in 751 innings pitched, giving him a lifetime ERA of 4.07. At the plate he was 35-for-206 (.170) with 3 home runs...the first two against Buster Narum and the third off Denny McLain.