Heading into the game in Pristina, Kosovo knew a draw against unbeaten Azerbaijan would be enough to wrap up top spot in Group D3 and boost their Euro qualifying hopes.
But they got off to a dream start, with Heerenveen midfielder Zeneli putting them ahead in only the second minute.
The 23-year-old scored twice more in the second half, with Dinamo Zagreb defender Amir Rrahmani also finding the net, as Kosovo completed a famous win.
Kosovo only claimed their first-ever competitive victory in September by beating the Faroe Islands, but have since won three more times in the Nations League with an incredible turnaround in form.
They will play at the League D playoffs next year, where Georgia, Belarus and Macedonia will be battling them for a ticket to the 2020 European Championship.
Kosovo is an Albanian-majority former province of Serbia that broke away from the then Yugoslav republic in a bloody war in 1998-1999 and declared independence a decade later.
Serbia refuses to recognise its former province.