Put into bat in Chittagong, Bangladesh had posted a strong 277-6 on the back of a string of top order cameos and Mushfiqur Rahim's unbeaten 67 down the order.
The 2-1 ODI series defeat was Bangladesh's first at home in two years.
England's chase got off to a good start before Nasir Hossain dismissed James Vince for 32.
Having added 63 runs with Vince, Billings added another 64 with Duckett, the duo helping themselves to fluent half-centuries in the process.
At 227-4 in the 41st over, England looked heavy favourites but Mashrafe Mortaza struck twice, dismissing England captain Jos Buttler (25) and Moeen Ali (one) in quick succession to inject fresh excitement into the contest.
Stokes, however, maintained his composure and hit two sixes and a boundary while running hard between the wickets to keep England on course.
Woakes proved a worthy partner as the duo added 42 runs in 32 balls to seal the victory.
Earlier, Buttler won the toss and decided to field but had to wait until the 19th over for a breakthrough.
Tamim Iqbal (45) and Imrul Kayes (46) added 80 runs for the opening stand before the guile of Adil Rashid (4-43), combined with some poor shot selection by the home batsmen, pegged back Bangladesh.
After Stokes sent back Kayes, Rashid dismissed Tamim, who became the first Bangladesh batsman to reach 5000 ODI runs, with a short and wide delivery.
Mahmudullah opened his account with a six off the leg-spinner but Rashid had the last laugh as the batsman drove a long-hop to short cover.
Sabbir Rahman (49) could not bring up his fifty either, becoming Rashid's third victim as Bangladesh lost momentum.
Mushfiqur and Mosaddek Hossain (38 not out) enjoyed a busy seventh wicket partnership that yielded 85 runs to take the hosts past the 275-mark.
The teams now face off in a two-test series starting in Chittagong on Oct 20.
(Reporting by Amlan Chakraborty in New Delhi; editing by Toby Davis)
