Can Barbarian’s Rage ability break the Charmed condition?

Can a Barbarian enter a Rage on his turn in combat if he’s charmed by Charm Person, or similar spells?

Last week during one of my games, our DM had a Cambion use a charm on the Barbarian, which he failed the saving throw and was ‘charmed’.

On his turn, he used his bonus action to enter a rage and ‘break’ the charm. The DM didn’t agree on this. Who is right? The Barbarian then continued on to attack with his normal action.