Is expiation obligatory upon one who falsely swore that he had done something, then promptly did it after the oath?

You erred by swearing a false oath. The duty was not to swear until you had acted, because lying is forbidden, and its prohibition intensifies with an oath. You are not required to offer expiation (kaffarah (definition)) according to the majority of scholars, because a false oath does not entail expiation. Expiation is only required for a binding oath (yameen mun'aqidah). You must repent to Allah and not return to such an act. Repentance is sufficient for you.