WASHINGTON — The FBI investigation that led to the resignation of David H. Petraeus as CIA director Friday began with a complaint several months ago about ‘‘harassing’’ e-mails sent by Paula Broadwell, Petraeus’ biographer, to an unidentified third person, a government official briefed on the case said Saturday.

When FBI agents following up on the complaint began to examine Broadwell’s e-mails, they discovered exchanges between her and Petraeus that revealed they were having an affair, said the official, who spoke of the investigation on the condition of anonymity.