WASHINGTON—The Senate rejected a one-month spending bill early Saturday, triggering the shutdown of many government services and setting off a partisan fight over who would bear the political consequences.

In a dramatic night of failed negotiations, Senate leaders extended the vote on the House’s short-term bill for more than an hour late Friday night as lawmakers attempted to strike a deal to avoid the shutdown. But efforts to agree to a stopgap measure floundered, leaving Congress with no way to avert the first major shutdown...