Chasing 217 to win, the Dutch were comfortably placed at 114-2 at one stage, before losing their next seven wickets for 71.
Final pair Paul van Meekeren (12) and Fred Klaassen (13) steadied the ship to leave the home side needing six off the last over.
Nepal impress despite losing in maiden ODI game
However, needing one run to tie and two to win off the last ball, Klaassen was run out by Nepal skipper Paras Khadka off his own bowling to give the sport's newest ODI side a thrilling win.
The series finished 1-1 after the Netherlands won the first match at the same venue by 55 runs on Wednesday.
Earlier, Sompal Kami, batting at eight, top-scored for Nepal with 61 off 46 balls with five boundaries and three sixes.
Khadka added 51 with six fours and two sixes as Nepal recovered from 88-5 to reach 216 all out in the 49th over.
I'll go to bed dreaming of getting Kohli out: Anderson
For the Dutch Wesley Barresi top-scored with 71 runs — an innings which featured eight boundaries.
Teenage leg-spinner Sandeep Lamichanne was the pick of the Nepal bowlers with 3-41.
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