On his own triumph, Hamilton added: "It's a great day for the United Kingdom and I hope that I've contributed in a positive way to that".
"Great work guys! I'm so happy for England as well", he said over the team radio, referring to the nation's 6-1 World Cup group win over Panama.
"That's what they said to me in the beginning of the season so I think they should do the same [to him]".
Sunday's victory was the 65th of Hamilton's F1 career but the Briton's first in France, which last hosted a race in 2008.
Vettel limped back to the pits to get a new nose piece while Bottas got a new tire for the one that was shredded by the nudge from the four-time former champion.
"In the end, the way I look at it with hindsight, I would have liked to have a worse start, because then it would have been more straightforward, I wouldn't be in that position".
"It was my mistake", conceded Vettel.
Mercedes Non-Executive Chairman Niki Lauda, however, was not impressed with Vettel nor his penalty, angrily calling it "an enormous mistake..." It is too little.
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Serbian Press Slams ‘Provocative’ Swiss Celebrations After 2-1 Defeat
The Argentinian FA and Croatian FA have also had disciplinary proceedings opened against them after crowd disturbances. But some Serbians may have been more angered by how Shaqiri celebrated the goal than the fact he scored it.
Vettel charged on the restart on lap six and moved swiftly through the pack, making it to 12th by the end of lap seven.
A better launch on his grippier Ultrasoft tyres allowed the Ferrari man to get alongside the two Mercedes' on the run to the first corner.
But Vettel couldn't close the gap with the Red Bulls and was passed by Raikkonen on fresh tires.
Hamilton had built a 20-second lead in the 33rd lap before pitting and emerging ahead of Verstappen and, briefly, behind Raikkonen until the Finn also changed his tyres.
The race got off to an eventful start as Vettel hit Valtteri Bottas on the first corner, and both cars sustained damage although they were able to continue.
Bottas didn't manage to recover from the back of the grid as well as Vettel - which makes you wish Vettel had been in the fight for the front throughout - and he finished in seventh place, behind the Haas of Kevin Magnussen.
McLaren's hard weekend also continued with Stoffel Vandoorne 12th and Fernando Alonso sounding increasingly frustrated on the radio and would, in fact, finish as the last classified runner after a late pit-stop to try and set the fastest lap.
A flying late lap by Leclerc lifted him to 10th for Sauber, pushing Esteban Ocon and his Force India team-mate Sergio Perez out of the shootout in 11th and 13th, either side of Nico Hulkenberg of Renault.