The law, which has re-ignited the deeply divisive debate about euthanasia, would also make "living wills" — drafted by people who do not want to be kept alive artificially if they are too ill to decide — legally binding on doctors.
"Sleep before death to avoid suffering," said MP Jean Leonetti, summing up the law he proposed.
Euthanasia is illegal in France but President Francois Hollande pledged in his 2012 presidential campaign to look into an issue that divides a country where heart-wrenching end-of-life stories continue to make headlines.
Polls show French people are overwhelmingly in favour (96 percent in a recent survey) of putting patients into a deep sleep if they are able to make the decision themselves.
This drops only slightly (to 88 percent) if the medical care team takes the decision because the patient is unable.
Eight out of 10 French people would even go further than the draft law, and legalise euthanasia.
A 2005 French law already legalises passive euthanasia, where treatment needed to maintain life is withheld or withdrawn.
But the proposed law goes further, allowing doctors to couple this with "deep and continuous sedation" for terminally ill patients.
The debate on euthanasia regularly opposes those who say the sanctity of life must be respected at all costs and those who believe terminally ill patients in unbearable pain must be allowed to die with dignity.
Health Minister Marisol Touraine has been steadfast in her refusal to legalise euthanasia and an amendment to the law allowing "medical assistance to die" was rejected.
Several deputies — notably from the Green party and the radical left — had vowed to abstain in the vote, saying it does not go far enough.
On the other side of the debate, around 20 deputies from the opposition right-wing UMP parties were poised to abstain because they believe the draft law goes too far towards allowing euthanasia.
Nevertheless, the bill as expected passed with a large majority and will then be presented to the Senate, or upper house of parliament.