Special counsel Robert Mueller’s office netted its first prison sentence on Tuesday and disclosed a memo showing it had been explicitly authorized to look beyond the presidential election, bringing into sharper focus its wide-ranging probe of Russian meddling in the 2016 election.

A lawyer who admitted to lying to Mr. Mueller’s investigators was sentenced to 30 days in prison by U.S. District Judge Amy Berman Jackson, who said the attorney “knew better” but had repeatedly misled investigators. “These are not mistakes, they...