On 6 June 1944 the invasion of Normandy started. During the night Luck was startled by the reports of paratroopers landing in his area, and establishing a bridgehead on the east side of the Orne River. A quick attack was launched by the II Battalion, and it succeeded in disturbing the paratrooper operations, and capturing some prisoners. Luck was, however, hampered by the strict orders not to engage in major operations unless cleared to do so by high command. As the day wore on, the defenders on the coast were smashed, while 21 Panzer Division remained mostly motionless, apart from an order at 4:30 a.m. directing other elements of the unit to move against the paratroopers of the British 6th Airborne Division and thus farther away from the coast.

Around 10:30 a.m. General Erich Marcks, commander of the German LXXXIV Corps to which 21st Panzer Division was attached, ordered the entire 21st to attack east of the Orne River. This was later countermanded from 7th Army high command, ordering only Luck's detachment to attack east of Orne, while the rest of the division should attack on the west side of the river. This naturally caused much confusion and further delayed the German response. Nevertheless, at 1700 p.m. Luck attempted to break through to the Orne river bridges at Bénouville with his Schützenpanzerwagen (armoured personnel carriers), but heavy fire from the warships supporting the British paratroopers, under Major John Howard, holding the bridges drove his forces back. Added to this, more British paratroopers landed in the rear area of the Regiment, forcing Luck's II Battalion to fall back to avoid getting surrounded. This battalion lost its commander on the morning of 7 June.

On the morning of the 9 June Luck's command was designated Kampfgruppe von Luck, and in addition to the elements of 125th Panzer Grenadier Regiment already under Luck's command it consisted of Panzer Reconnaissance Battalion 21, 4th Company, Panzer Regiment 22, three batteries from Major Alfred Becker's Assault-Gun Battalion 200 and one company from Antitank Battalion 220 (88mm guns). With this force Luck was again tasked with assaulting the Orne bridges, and recapturing them from the British paratroopers. Starting one hour before dawn to avoid the worst of the British naval and aerial support, the Kampfgruppe advanced on the village of Ranville, dislodging the enemy there, but it could not penetrate the British lines to reach the bridges. The British paratroopers had been reinforced by the British 51st (Highland) Division on the evening of 8 June.

On 12 June Kampfgruppe von Luck, now further enlarged with an addition of a brigade of Nebelwerfers, successfully reclaimed the village of Sainte-Honorine, lying on an important hill overlooking the invasion beaches. A furious counterattack by a Canadian Division resulted in the Germans having to withdraw again, after fierce hand to hand fighting. After this final attack had been repulsed, Luck determined that the British bridgehead could not be eliminated, but due to the counterattacks launched by Kampfgruppe von Luck, the British/Canadian forces stopped any further advance in the sector, preferring to lay mines and dig themselves in.[19] Apart from a failed German attack on 15 June, the sector was relatively quiet for the next two weeks.