England posted a meagre 128-9 with the bat, their lowest total against Sri Lanka in the format, as pace bowler Dushmantha Chameera bagged a five-wicket haul, but all-rounders Jacks (3-14) and Bethell (4-11) produced their best bowling performances in the format to restrict the hosts to 116.
"We've shown we can adapt to surfaces... the way the spinners bowled, 16 overs against Sri Lanka in their own conditions was an awesome effort," England captain Harry Brook said.
Ben Duckett was dismissed for a golden duck by Chameera, and Bethell, Tom Banton and Brook fell cheaply, leaving England struggling at 34-4 in the sixth over.
Jos Buttler (25) was beaten by Maheesh Theekshana's sharp turn off Kandy's spin-friendly track, which found the gap between the England opener's bat and pad to hit the stumps.
Player of the Match and Series Sam Curran made a battling 58 to lift England to a respectable total, and Luke Wood removed Kamil Mishara for a duck in the first over. The spinners then took over, with Liam Dawson getting Pathum Nissanka caught at cover for 23.
Adil Rashid bowled Pavan Rathnayake, before Jacks worked through Sri Lanka's middle order, dismissing Kusal Mendis (26), Kamindu Mendis (14) and Janith Liyanage (17) to reduce Sri Lanka to 97-6 in the 16th over.
Bethell took three wickets in the 18th over, and with Sri Lanka needing 14 to win from the last over, Theekshana's attempt to hit Bethell for six got him caught as the hosts were all out with three balls remaining.
"Great win to be part of, the way the boys kept fighting to the end... the World Cup is here, so when we have the opportunity to play here we have the experiences to carry over," Brook said.
England, twice world champions in the format, will fly to Mumbai to begin their T20 World Cup campaign against Nepal in Group C on Sunday, the same day Sri Lanka face Ireland at Colombo in Group B.