Lukaku scored twice, while Barkley also found the net, as the Toffees defeated Newcastle United 3-2 in the Premier League on Monday night.
"I love playing with Rom," the 19-year-old told the club's official website. "We link up well and I can run beyond him and I know he'll find me.
"We've got a good relationship on the field and it's good having a top player like him in the side."
Monday's victory over Newcastle saw Everton continue their run as the only unbeaten side in the Premier League.