WASHINGTON—Deputy Attorney General Rod Rosenstein argued his first case at the Supreme Court on Monday, a criminal sentencing dispute from New Mexico that has no connection to Russia’s alleged interference in the 2016 presidential election.

While something less than the integrity of American democracy hung in the balance, the case satisfied Mr. Rosenstein’s professional ambition to argue before the high court. That is a rare privilege for most attorneys, and perhaps one of the few perks of a job that, thanks to his role supervising...