WASHINGTON—The Senate on Monday approved a short-term extension of a batch of expired domestic surveillance powers that lapsed over the weekend, choosing to delay consideration of a host of privacy changes to pivot attention more quickly to the rapidly escalating coronavirus crisis.

The move came after Republicans engaged in a public face-off last week over whether to vote on a House-passed measure that included a number of changes to the Foreign Intelligence Surveillance Act—which has attracted renewed scrutiny in recent...