The Azzurri were unable to repeat the heroics of their 22-21 win in Rome in last year's tournament.
But having kept in touch with Les Bleus for most of the match, Brunel - who was taking charge of his first match as Italy coach - says it is only the finishing ability that is holding his side back.
"In the first half, we kept a lot of possession and took the initiative, and dominated territory, and were only 15-6 down at halftime," said Brunel.
"I think we enjoyed 75 per cent of territory and 65 per cent of possession in the first-half.
"We also had three or four chances to score a try but we never managed it, as France did.
"That shows their value, they were able to turn everything around with very few chances. That's the difference."
With captain Sergio Parisse was as powerful and influential as ever, Italy were more than a match for the French in the first half before the hosts' killer touch saw them pull clear.
And Brunel says there is a lot for them to build on heading into their next fixture against England in Rome.
"To know that to dominate is not to win," he added. "We put the French defence under pressure but the final gesture just got away from us, but the spirit was there and that's the most important thing.
"We showed a good spirit on the pitch, always trying to attack and put the French on the backfoot. It's sometimes difficult to find the words to explain the missed passes and everything else that went wrong.
"But we're on an interesting path. We want to continue advancing."