"Four other navy personnel and crew were injured," Addax said in a statement. No group has so far claimed responsibility for the attack.
The incident did not affect the operations or the technical infrastructure of the company, Addax said.
The navy personnel and vessel have been taken to Bonny Island where the injured are receiving medical treatment, the company said.
Jean Claude Gandur, the Addax chairman and CEO said in a company press release: "We are deeply saddened by this incident and our thoughts and prayers are with the victim's family and those of the injured personnel."
Oil companies and their personnel are being increasingly targeted by armed groups who have stepped up kidnappings and sabotage in the region in the past two years.
Addax Petroleum Nigeria is an affiliate of Geneva-based Addax and Oryx Group, an international oil and gas exploration company focused on West Africa and the Middle East.
Last month, Nigeria's most prominent armed group, the Movement for the Emancipation of Niger Delta (MEND), attacked an oil pipeline run by Royal Dutch Shell in the delta and claimed to have killed 11 soldiers.
Attacks have affected Nigeria's daily production, which has been cut by about a quarter.
Nigeria was overtaken as Africa's largest producer by Angola in April, according to Organisation of Petroleum Exporting Countries' (Opec) figures.