Samantha Murray finished second to pick up the silver medal and maintain Great Britain's record of having won a Modern Pentathlon medal at every Games since the women's event was introduced to the programme in Sydney 2000.
Brazil's Yane Marques held on to claim the bronze medal.
Asadauskaite said afterwards: 'I am immensely excited. I had a great deal of support from the people of Lithuania. It is only a small country, so this means the world to them.
'I felt very excited and emotional when the crowd cheered for me at the end of the event. But I think they were also cheering for all the other athletes.'