The 32-year-old headlined on BT Sport and fought out of the home corner at the Telford International Arena after teaming up with Frank Warren and Queensbury.
Having won British, Commonwealth and European titles in the past year, Cunningham’s stunning renaissance continued with a dominant win over his French opponent.
He knocked down Le Couviour three times with body punches before stopping him in the sixth.
Cunningham told BT Sport: “I’m over the moon bar one round in which I switched off.
"It was unknown what Le Couviour was all about but I used my jab and broke him down with the body shots.
"Onwards and upwards now, whatever makes sense. Obviously I just want to push towards a world title.”