Šenavičius, 35, had served in the mission since August 2012. He joined the Klaipėda territorial customs in 2000.
According to the press release, EULEX officers came under fire during their service near the city of Zvecan in northern Kosovo close to the Serbian border.
EULEX website said the attack took place at about 7:30 AM.
"The shooters at the EULEX officers are yet unknown. Two more officers were injured in the attack, they are undergoing treatment in hospital. EULEX has launched an investigation. KFOR troops have been dispatched to the scene," according to the department.
"Two EULEX vehicles, carrying six staff members, on regular rotation from Gate 1 came under fire from unknown persons in the vicinity of Municipality Zvecan. The incident resulted in the death of one EULEX staff member," EULEX said on its website.
"Task Force Mitrovica was immediately activated and a murder investigation is ongoing. The area has been sealed off," the mission said in its press release.
EULEX is a civilian mission.
Foreign Minister Linas Linkevičius has condemned the use of violence against EU personnel.
"Linkevičius expressed his condolences over the death of Lithuanian customs officer Audrius Šenavičius in northern Kosovo to the bereaved family and strongly condemned the use of violence against the staff of the peaceful European Union Rule of Law Mission," the Foreign Ministry said in a press release.
"I send my deepest condolences over the death of the officer to the bereaved family and wish them strength at this time of grief," the ministry cited Linkevičius as saying.
