Victory saw England complete their first 5-0 series sweep of Australia in any format in more than 140 years of international men's matches between the arch-rivals.
Set just 206 to win, England slumped to 50 for five and 114 for eight.
They were still 11 runs shy of victory at 195 for nine when Adil Rashid (20), who had helped Buttler add 81, was dismissed to leave England 195 for nine.
But Buttler responded by hitting part-time seamer Marcus Stoinis for six to complete a hundred off 117 balls -- the first time the usually rapid-scoring wicketkeeper-batsman had faced more than 100 balls at this level.
Last man Jake Ball held his nerve and Buttler won the match with nine balls to spare when he hit Stoinis for four.
It was a win celebrated by raucous cheers from 23,500 capacity sun-drenched crowd, some of whom had previously enjoyed the goals the England football team scored in a 6-1 World Cup rout of Panama.
Fast bowler Billy Stanlake's opening spell of three wickets for 18 runs had given Australia a shot of a first international win under coach Justin Langer, appointed after Darren Lehmann resigned in the fall-out from the Cape Town ball-tampering scandal in March.
It took England just four balls to lose their first wicket when the in-form Jason Roy, fresh from his 101 in the fourth ODI at Chester-le-Street on Thursday, was bowled attempting a big hit off left-arm spinner Ashton Agar.
Stanlake then bowled Jonny Bairstow (12) and had Test skipper Joe Root (one) well caught at slip by Shaun Marsh.
When Stanlake bowled England captain Eoin Morgan for a duck, the hosts were 27 for four.
England were 66 for five off 16 overs when 'lunch' was taken.
Buttler (27 not out) and Moeen Ali (three not out) were then at the crease, but but Ali (16) fell to Stoinis soon afterwards.
And England were 114 for eight in the 30th over when Kane Richardson struck twice in two balls to remove ODI debutant Sam Curran and Liam Plunkett.
Rashid survived the hat-trick, however, and then helped take Buttler to the brink of victory before he was well caught by Stanlake, at fine leg, off Stoinis.
Earlier, off-spinner Ali took an ODI best four for 46 as England, the number one-ranked side in this format, dismissed world champions Australia, who won the toss, for 205 inside 35 overs.
Travis Head (56) was the only Australia batsman to pass fifty, with D'Arcy Short stranded on 47 not out.
Together with opening partner Aaron Finch, he put on 60 inside seven overs.
It was then that Ali took two wickets for no runs in three balls as spin again proved to be Australia's Achilles heel.
With his third ball of the match, Ali had Finch (22) under-edging a pull onto his stumps.
Two balls later, Stoinis was out for nought when a mistimed sweep lobbed gently to Ball at short fine leg.
Head compiled a 36-ball fifty, including eight fours but again fell soon afterwards when he chipped fast bowler Liam Plunkett to Morgan at short midwicket.
And 90 for three became 97 for four when Marsh, who had made two hundreds this series, exited for eight, stumped by Buttler off Ali.
Wicket-keeper Buttler was involved again as Australia slumped to 100 for five when, reacting quickly to Australia captain Tim Paine's nudge, he threw down the stumps at the bowler's end to run out his opposing gloveman for one.
Surrey left-arm seamer Curran then followed Ali in taking two for none in three balls to leave Australia 159 for seven.
Alex Carey had made 44 when he was caught behind trying to steer the ball to third man.
Two balls later Agar was out for a bizarre duck when bowled shouldering arms to a straight Curran delivery that smashed into his middle and off stumps.
ashton agarBilly StanlakeEngland vs Australia 2018Joe RootJonny BairstowJos ButtlerMoeen Alishaun marsh
First Published: June 24, 2018, 11:16 PM IST