The visitors produced a powerful display but were ill-disciplined at times, enabling scrumhalf Maxime Machenaud, who scored all France's points from penalties, to keep his team in contention throughout.
First-half tries by Danny Care and Dan Cole and one more after the break by Anthony Watson, the rest of the points coming from Owen Farrell's boot, proved enough to give England the win they craved.
On a crisp evening on the outskirts of Paris, they sealed their 13th all-time clean sweep, two more than next-best Wales, bouncing back in style from last year's early World Cup exit to hand new Australian coach Eddie Jones a perfect debut campaign.
England, who made sure of winning the tournament for the first time since 2011 with a win over Wales last week, lost Dylan Hartley for the closing stages.
The England captain left the pitch on a stretcher after a clash with substitute prop Uini Atonio.
France, also rebuilding with a new coach in former Toulouse boss Guy Noves, started the tournament in promising fashion with victories over Italy and Ireland before defeats by Wales and Scotland ended their title hopes.
The two sides had traded penalties when Care escaped from a ruck and sprinted unchallenged to touch down between the posts, handing Farrell an easy conversion.
MARLER ON
France flyhalf Francois Trinh-Duc then limped off to be replaced by Jules Plisson.
Forward pressure brought England's second try from Cole and the visitors would have been in control had they not let Machenaud showcase his kicking ability.
England's defence held tight but did survive a few scares, notably some fine runs from speedy winger Virimi Vakatawa and fullback Scott Spedding.
Jones made two changes from the side that started against Wales, bringing in Care for Ben Youngs at scrumhalf and Mako Vunipola for Joe Marler at loosehead prop.
Marler was sent for the second half with England 17-12 in front.
Jones' men struggled at times after the break before winger Watson picked up an astute Youngs kick to dive over in the corner for their third try.
The champions could always rely on the raw power of man-of-the-match Billy Vunipola, once again outstanding at number eight, and on the accuracy of Farrell.
Farrell wrapped it up with yet another penalty in the dying moments as England fans sang 'Swing low, sweet chariot' to celebrate their success.
(Reporting by Patrick Vignal; Editing by Tony Jimenez)
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