Tuilagi scored the hosts' only try at Twickenham Stadium as they maintained their winning start to the tournament.
"We're relieved we got through it," he told BBC Sport. "The boys worked hard out there - it was a tough game.
"The French played really well and I think our defence was really good. It was a bit messy at times but we scrambled back. We were happy to get the win."
England return to action on March 10 when they entertain Italy.