WASHINGTON—A record 3.28 million workers applied for unemployment benefits last week as the new coronavirus hit the U.S. economy, marking an abrupt end to the nation’s historic, decadelong run of job growth.

The number of Americans filing for claims was nearly five times the previous record high. The surge was for the week ended March 21 and could rise further. Pennsylvania, Ohio and California were among 10 states reporting more than 100,000 claims, leaving unemployment systems overloaded.

Millions...