The victory was Australian coach Eddie Jones’ fifth straight win since taking over from Stuart Lancaster, the English having seen off Scotland (15-9), Italy (40-9), Ireland (21-10) and Wales (25-21).
It was England’s first Grand Slam since their World Cup-winning year of 2003, when Jones was in charge of losing finalists Australia.
But it was not easy going at the Stade de France for England, who scored three tries through Danny Care, Dan Cole and Anthony Watson with Owen Farrell contributing 16 points with the boot.
A rash of ruck infringements allowed France to keep in the game through scrum-half Maxime Machenaud, who booted seven penalties in a flawless kicking display.
Machenaud got the scoreboard rolling after Maro Itoje was pinged at a ruck, but Farrell responded immediately when France skipper Guilhem Guirado was penalised for a dangerous tackle.
France delighted a packed stadium by running the ball with intent, Scott Spedding and Virimi Vakatawa, born in South Africa and Fiji respectively, both splitting the English defence only for final passes to go amiss.
Livewire scrum-half Care, who dictated a fast pace, showed up some lax ruck defence by stepping straight through a gap left by lumbering prop Jefferson Poirot and crossing for a well-taken try, converted by Farrell.
France suffered a blow when Francois Trinh-Duc limped off after 13 minutes to be replaced by Jules Plisson at fly-half.
Spedding broke through again, with France finally showing their attacking muster after generally failing to fire in tight wins over Italy (23-21) and Ireland (10-9) and defeats by Wales (19-10) and Scotland (29-18).
Machenaud booted his second penalty, but England again came firing back up the pitch, prop Cole bundling over the line for a try Farrell again converted.
England, with Billy Vunipola and flanker James Haskell to the fore, denied wave after wave of home attack, Machenaud keeping France in the game with a third penalty after another ruck infringement.
It was then France’s turn to defend and they did so with aplomb as Care marshalled his team from left to right, before two knock-ons undid English efforts.
Machenaud knocked over a fourth penalty to leave the score 17-12 at half-time with Farrell missing his second effort.
France opened the second period with an early charge by Vakatawa, the winger hauled down by opposite number Jack Nowell with the line abegging, Machenaud kicking a fifth penalty to bring the home side within two points.
The tit-for-tat continued as Farrell banged over his own penalty as Guirado strayed offside before Machenaud booted his sixth.
Then came Watson’s try after a brilliant break by Care’s replacement Ben Youngs, who put a grubber kick into the corner rather than risk a pass, the Bath winger picking up and barging through Wesley Fofana.
From the restart, England again infringed, Machenaud making no mistake from the tee.
France made a raft of changes to the front five, temporarily giving them the upper hand at the scrum, although their line-out was creaking and gifted England vital possession.
Farrell calmed nerves with his third and fourth penalties to ensure England the Grand Slam, having contrived to botch four other occasions to nab the trophy since the Five Nations became Six in 2000.
