Supporters gathered outside Parliament wept and hugged each other as the result was announced.
Campaign group Dignity in Dying said the vote was a “historic step towards greater choice and protection for dying people”.
Dame Esther Rantzen, one of the most high-profile campaigners for assisted dying, said she was “absolutely thrilled”.
The broadcaster, who has terminal lung cancer, said any change in the law would probably come too late to affect her personally.
But she said “future generations will be spared the ordeals we have to suffer at the moment” if the bill becomes law.
Labour MP Kim Leadbeater, who put forward the bill, told the BBC she was “a bit overwhelmed” following the vote and it meant “a huge amount” to be able to tell campaigners the bill had passed its first parliamentary hurdle.
However, Conservative Danny Kruger, a leading opponent of the bill, said it could be defeated at a later stage if MPs’ concerns were not properly addressed.
He said many of his colleagues believed the bill was “very dangerous” and he hoped that if safeguards in the legislation were not strengthened they would choose to vote against it in the future.
