But his championship rival Sebastian Vettel made an error on the opening lap that forced him nearly to the back of the field and he could manage only fifth.
Kiwi Brendon Hartley benefited from the carnage, moving up to 13th after starting last on the grid.
Vettel, at the back after a pit stop to fix the front wing damaged in the crash with Bottas made light work of the backmarkers, starting with a light tap as he passed Fernando Alonso's McLaren at Turn Three at the re-start, which nudged the Spaniard into a spin. The Red Bull's pitted on laps 26 and 28 and Vettel took fourth from the Australian though the stops.
Hamilton extended his record run of points-scoring finishes to 33 successive races.
"Great work guys, so happy for England as well it is a attractive Sunday everyone", he said over his team radio. "I was hit in Turn 1, got a puncture, that made quite a bit of damage in the floor, which made it quite hard to comeback through, but at least we got some points".
Pivotal to the turnaround in performance was the introduction of Mercedes' 2.1 spec engine, with the Silver Arrows simply too powerful for their rivals after around-the-clock work at their Brixworth factory.
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However while in Canada they had not employed their new, upgraded engine, in France they had the unit and it had proved to be formidable. It definitely was a positive and the guys did a great job on the engine side of things and the development part.
It wouldn't be so bad if we have to take the spec 1 again.
Hamilton moved 14 points clear at the top of the standings after his 65th career win. Only Michael Schumacher has more victories with 91.
Other collisions on a chaotic first lap at Le Castellet saw two of the three French drivers retire, with Force India's Esteban Ocon and Toro Rosso's Pierre Gasly the unlucky casualties.
Both drivers continued on, with Vettel recovering to fifth while Bottas came home seventh.
Vettel was hit with a five-second penalty, but it had little effect on his comeback drive from 17th to fifth. Despite his struggles, Hamilton managed to finish just over a tenth of a second faster than his teammate. "Mistakes happen in life, it happens to the best of us as you can see today so to be honest, it just makes me angry".
The 90-minute opening session was red-flagged with a minute remaining when Ericsson lost control at the Beausset corner, spun and hit the barriers hard, with the Sauber vehicle then catching fire.