Sounds better "He acabado(terminado) con mis exámenes".. ;)..you are the person who make the action and not the "exámenes".

just a "footnote", "terminado" will be more used in Chile and I'd say South America. "Acabado" is perfect but not very used in Chile at least.

From a strict grammar point of view both sentences are fine but using "terminado" is absolutely more used than acabado. However, considering how people speak, we'd use the active voice for this sentence as the two other comments said. In addition, You can find some little differences as "...con mis examenes" and "... mis examenes" (without "con"). That depends on the country but eveybody will understand both ways.

Sounds better: "He acabado (terminado) mis exámenes". "Mis exámenes" is COD, direct complement. What has been finished? My exams. Like "I've finished my homeworks".