In 1939 Goldstine began a teaching career at the University of Michigan, until the United States' entry into World War II, when he joined the U.S. Army.

As a result of the United States' entering World War II, Goldstine left the University of Michigan where he was a professor in July, 1942 to enlist in the Army.

He was commissioned a lieutenant and worked as an ordnance mathematician calculating firing tables at the Ballistic Research Laboratory (BRL) at Aberdeen Proving Ground, Maryland.

In spite of disappointment that ENIAC had not contributed to the war effort, interest remained strong in the Army to develop an electronic computer.

Goldstine, Mauchly, J. Presper Eckert and Arthur Burks began to study the development of the new machine in the hopes of correcting the deficiencies of the ENIAC.

As a result of his conversations with Goldstine, von Neumann joined the study group and wrote a memo called First Draft of a Report on the EDVAC.

The fact that Eckert and Mauchly, the actual inventors and designers of the ENIAC, were not named as co-authors created resentment that led to the group's dissolution at the end of the war.

After World War II Goldstine joined von Neumann and Burks at the Institute for Advanced Study at Princeton, where they built a computer referred to as the IAS machine.

Goldstine went on to become the founding director of the Mathematical Sciences Department at IBM's Watson Research Center in Yorktown Heights, New York.

In retirement Goldstine became executive director of the American Philosophical Society in Philadelphia between 1985 and 1997, in which capacity he was able to attract many prestigious visitors and speakers.