A bill that would renew extended-unemployment benefits cleared a hurdle in the Senate Tuesday, ending weeks of stalemate that had led to the expiration of federal aid to millions of jobless Americans amid a debate on the legislation's cost.

In a vote chiefly along party lines that highlighted concerns over how to pay for the bill, the Senate broke a filibuster led by Republicans and set the stage for a full Senate vote on the legislation, which could come as soon as Wednesday.

The...