Nweke told a news conference: "Last night our team of 13 scientists were able to conclusively identify the case of avian influenza in a 22-year-old female who died in Lagos."
Abdulsalam Nasidi, a bird flu expert at the health ministry, said the three inconclusive cases were the mother of the confirmed case, a poultry worker in Taraba state and one person in Borno state.
The woman was one of 14 people, three of whom died, from whom samples were taken for tests that were concluded on Tuesday.
Samples are being sent to overseas laboratories for confirmation.
Nigeria is one of three countries regarded by experts as the weakest areas in the global attempt to stem the spread of the virus.
Bird flu has killed at least 164 people since it re-emerged in Asia in 2003, according to the most recent figures from the World Health Organisation. The toll includes 11 in Egypt.