WASHINGTON — Pressed by shooting victims and relatives of Americans slain in gun violence, the Senate on Thursday voted to begin an emotionally and politically charged debate on gun safety proposals as advocates of new laws overcame a Republican filibuster threat.

The strong majority in favor of considering legislation that would expand background checks and increase the penalties for illegal gun sales reflected the power of a lobbying campaign by parents of students killed in Newtown, Conn., and by others who persuaded reluctant lawmakers to back them in an initial fight that looked lost just last week. The vote was 68 to 31.

“It’s remarkable,” said Senator Christopher S. Murphy, a Democrat from Connecticut whose nascent Senate career has been devoted to gun safety. “You can’t turn a corner in the Capitol this week without meeting a family of a gun violence victim. It’s hard to say no to these families.”

But the victory could be short-lived. The vote in no way guaranteed passage of the gun measure; some Republicans and Democrats who voted for this initial step made clear they are not committed to supporting any final measure, even if they agreed to allow the debate.