Brunel: France can compete with the best
Maxime Machenaud (4) and Lionel Beauxis landed penalties for France, who were also awarded a penalty try.
England's Owen Farrell (2) and Elliot Daly responded with their own penalties before Jonny May crossed for a try Farrell converted to set up a frantic finish to what had been a dour, defence-driven and often mistake-riddled encounter at the Stade de France.
The victory was France's second after they saw off Italy last week to break a losing streak going back 11 months.
Allied to that has been France's further dip in popularity after a controversial late-night, alcohol-fuelled drinking session following defeat in Scotland that saw eight players sanctioned by Brunel.
"France has gone through a tough time," the 64-year-old Brunel said.
"I believe we can come very close to the best. Against Ireland (lost 13-15 in the last minute), we were there, we lacked just a little bit at the end.
"Against both Ireland and England, we've shown we can go up against them and we will continue to do so."
Brunel added: "There were a lot of good things today.
"Our defence was remarkable. The line-out was a bit wobbly and the scrum as well. But there was a lot of energy and a lot of will from the players.
"There is a lot to be happy about. We knew that the physical challenge of the game would be a key factor. We controlled the ball and that upset them because they're not used to it."
Skipper Guilhem Guirado said he was "proud" of the team's spirit.
"We've been criticised a lot, but we never lost faith in the squad," the hooker said.
"It was a match of very high intensity but that's what sport's all about. It's always good to win against England, it doesn't happen too often. We're going to enjoy the victory."
England coach Jones saw his team concede a massive 16 penalties, with Anthony Watson also yellow carded for the high tackle on Benjamin Fall that saw South African referee Jaco Peyper award the penalty try to France.
"Obviously I'm very disappointed. We didn't take the opportunity to score points," the Australian said.
"The breakdown again caused us trouble. We improved in that area, but not to the extent we needed to. We just didn't learn quick enough. There's no lack of effort."
Jones, who has now led England to 24 victories in 27 matches since taking over after the 2015 Rugby World Cup, added: "Any team that's developing goes through a period where the game doesn't like you.
"We're in the loser's chair and it's not a happy place, but I don't think we should get too melodramatic about it.
"I was pleased with the effort today, we were in a position to win the game," he said, adding that any team was "fallible and has weakness".
Jones said that "some teams are outplaying us in certain areas". "The breakdown has definitely become more contestable. The contest has increased enormously, we have to find a way to cope with that. It's not going to be easy."