Well, its a bit complicated. .
When the first computers were being developed there was no "Department of Defense" it was the "War Department". If you really mean DOD, getti…ng an answer for first computer will be difficult as the WD had developed many earlier. .
Assuming you meant "War Department", it is still complicated as they were funding 3 separate computing machine development projects throughout most of the duration of the war that for security reasons knew little or nothing about each other. Those projects were at the following sites (listed in order of start date): .
MIT had begun a series of projects to build a wide variety of both analog and digital computing devices in 1928. By the mid-1930s the War Department was funding most of this work and had classified it. .
Harvard University shortly after the attack on Pearl Harbor Howard Aiken convinced the Navy to sign a contract to construct and operate the ASCC (later called the Harvard Mark I) that he had designed. IBM was selected to build it. This project was unclassified and was publicly announced in 1944. .
University of Pennsylvania in early 1943 signed a contract with the Army to design and build ENIAC. This project was classified. The only one of these to produce a functional programmable computer was the Harvard team, in 1944. Yes it was electromechanical and slow, but it was largely automatic, completed before the end of the war, and contributed to the solutions of many war related problems (including an early analysis of the implosion assembly of a supercritical mass for the Manhattan Project). MIT's work produced machines that contributed to the war effort, but they required step by step operator control. The University of Pennsylvania work on ENIAC was not completed until late 1945, after the war had ended. (MORE)