Grabsky was born on 3 April 1923 in Kyiv to a working-class Jewish family. On 25 June 1941, following the beginning of Operation Barbarossa, he volunteered for the Red Army, and originally served in the infantry. After being wounded at the Battle of Stalingrad, he was sent to join a tank crew. As commander of the 52nd Tank Guards Brigade, he distinguished himself in the Autumn of 1943, during the Battle of Kiev. In late September 1943, Soviet troops established a bridgehead in the Lyutizh, on the right bank of the Dnieper. Just before dawn on 4 November 1943, the 52nd Tank Guards Brigade's tanks landed on the bridgehead, via ferry. During an attack on enemy positions, Grabsky was in the vanguard, destroying one Tiger tank and as a result, helping the 52nd Tank Guards Brigade to cross the Dnieper.