BUCHAREST, Romania—Not long ago, the prosecutors at Romania’s anticorruption office rarely took on a bribery suspect bigger than a schoolteacher or a small-town cop. Often, the criminal gift was a pig, a lamb or a turkey.

These days, the National Anticorruption Directorate is investigating some 2,100 abuse-of-office cases, many against top politicians. Senators, parliamentarians and ministers have had to stand trial, and alleged damages investigated last year totaled €1 billion ($1.06 billion). Whistleblowers have started...