Kagawa was born in Hiroshima. He graduated from the 46th class of the Imperial Japanese Naval Academy, ranked 53rd out of 124 cadets. He served as midshipman on the cruiser Azuma, and as ensign on the cruiser Aso and destroyers Ariake and Susuki. After his promotion to lieutenant, and tours of duty aboard the battleship Fusō and cruiser Natori, Kagawa attended advanced navigational training courses at the Naval War College (Japan). On graduation and after promotion to lieutenant commander in 1930, he was assigned as chief navigator on a number of ships, including the cruisers Yūbari and Takao, and battleship Fusō. He received his first command — the destroyer Fubuki — on 16 November 1936. He then was captain of the destroyer Shionome and executive officer of the cruiser Ōi.
Kagawa was promoted to captain on 15 October 1941, and after a series of staff positions, was appointed commander of the 31st Destroyer Group (Desdiv 31) on 12 February 1943. Kagawa's group consisted of the destroyers Ōnami (flagship), Makinami, Amagiri, Yūgiri and Uzuki. During the Battle of Cape St. George on 26 November 1943, Onami was sunk and Kagawa was killed in action. He was posthumously promoted to rear admiral.