Mr. Hagel said “most DoD civilians” would be exempted from the furloughs and would return to work next week because Pentagon and Justice Department lawyers interpreted a stopgap budget measure signed into law last week by President Obama, which guaranteed pay for service members, to also apply to a larger number of civilian workers.

That action came as the House, in a rare Saturday session, voted unanimously to guarantee that all federal workers will receive back pay once the government shutdown ends, offering a promise of relief, if not an actual rescue, to more than one million government employees either furloughed or working without pay.