Cossack was laid down at the High Walker Yard of Vickers Armstrong at Newcastle-on-Tyne on 9 June 1936, launched on 8 June 1937 by Mrs. S. V. Goodall, commissioned on 7 June 1938 and completed on 14 June 1938.
Cossack's first action was on February 16, 1940. This was the Altmark Incident in Jøssingfjord, Norway which resulted in the freeing of the Admiral Graf Spee's prisoners who were being held aboard the supply ship Altmark and the death of seven members of the crew of the Altmark.
Shortly afterwards Cossack participated in the Second Battle of Narvik in (April 1940). In late 1940, she was part of the force which was assigned to hunt for a German surface raider that had been reported as breaking out into the North Atlantic. The force consisted of the battlecruiser HMS Hood, light cruiser Edinburgh, and destroyers HMS Electra, Echo, Escapade, and Cossack. The report turned out to have been false, and after spending a week at sea, including Christmas Day, she returned to port on New Years Eve.
In May 1941, she participated in the pursuit and destruction of the Bismarck. While escorting Convoy WS-8B to the Middle East, Cossack and 4 other destroyers broke off on 26 May, and headed towards the area where Bismarck had been reported. They found her that evening, and made several torpedo attacks in the evening and into the next morning. No hits were scored, but they kept her gunners from getting any sleep, making it easier for the battleships to attack her the next morning.
On 24 October 1941, Cossack was escorting a convoy from Gibraltar to the United Kingdom when she was struck by a single torpedo fired by the German submarine U-563. She was taken in tow by a tug from Gibraltar on 25 October, but the weather worsened and the tow was slipped on 26 October. Cossack sank in the Atlantic west of Gibraltar on 27 October 1941. 159 of her crew were lost.
HMS Cossack was named after the Cossack people who lived in the steppes of Ukraine and Southern Russia.