Drafted by the Cincinnati Reds in the 1st round of the 1987 MLB amateur draft, Armstrong made his big-league debut on June 21, 1988, and appeared in his final game on April 17, 1994. A torn rotator cuff helped end his career at the age of 27. Armstrong was also a member of the inaugural Florida Marlins team in 1993.
Armstrong had the best year of his career with the Cincinnati Reds in 1990. He won eight of his first nine starts and was the National League's starting pitcher in the All-Star Game. He tailed off, however, ending the year with a record of 12–9 and a 3.42 ERA. He relieved Scott Scudder in Game 2 of the 1990 World Series with the Reds trailing, 4–3. Armstrong pitched three scoreless innings against the Oakland Athletics, and the Reds won the game in the 10th inning.
After his career 1990 season, he spent one more year with the Reds, then spent one season each with the Indians, Marlins, and Rangers before retiring.
^"Armstrong Disarms Mets", The Record (Bergen County), May 4, 1990. "OK, let's get the obvious out of the way. Born in Englewood and a star at Neptune High School who went on to pitch at Rider College and the University of Oklahoma, 6-foot-5, 220-pound Cincinnati right-hander Jack Armstrong fulfills the qualifications for the obvious nickname, All-American Boy, like the fictional character of the same name."