Oesau, once described as "the toughest fighter pilot in the Luftwaffe", was from the start of the war amongst the top-scorers. He enlisted in the army in 1933 and served in an artillery regiment. By 1934 he had become a Fahnenjunker and was undertaking flying training.

On completing his flying training his first posting was to Jagdgeschwader "Richthofen". Lt. Oesau was one of the first fighter pilots to join J/88 in Spain in April 1938. Here he served with 3. J/88 and gained nine victories in 130 combat missions.

After leaving I./JG 21 he was posted with the rank of Oberleutnant to lead 7./JG 51. On 18 August 1940, he became the fifth Luftwaffe pilot to reach 20 victories, bringing him the award of the Ritterkreuz.

On 25 August 1940, Hauptmann Oesau was appointed Gruppenkommandeur of III./JG 51. A transfer followed on 11 November 1940, when Oesau was appointed Gruppenkommandeur of III./JG 3. After reaching 40th victory on 5 February 1941 on the channel front the next day, on 6 February, he was awarded the Eichenlaub.

Thearfter "Gulle" led this Gruppe to the Eastern Front where he recorded his first on the third day of the invasion, a I-18 (MiG-1) for his 43rd confirmed victory. He recorded his 50th victory on 30 June. And on 8 July he would amassed his 60 to 63rd abschusse in two separate missions. During the first he bagged three SB-2 bombers within four minutes. In his second mission in the later afternoon he shot down another SB-2.

However during this clash his machine was hit with a large caliber shell in the rear fuselage, resulted in 20% damage to his Bf 109 F-2, W.Nr. 9624.

Three days later Hptm. Oesau reached 70 confirmed victories. And a week after that his 80th adversary fell before his guns. This was a feat that resulted in the award of the coveted Schwertern as the third recipient in the Wehrmacht.

On 28 July 1941, he was recalled to the Western Front to take over the command of JG 2. By this time, fighting little more than a month on the Eastern front he had amassed at total of 44 confirmed victories.