Australia, who suffered their first series whitewash against the hosts in the three Tests, chased down 228 in 46.5 overs at the R. Premadasa Stadium to lead the five-match ODI series 1-0.
Pacers James Faulkner and Mitchell Starc, who became the fastest to reach 100 wickets in his 52nd match, also played their part with seven wickets between them after Australia elected to bowl.
"Nice to be on this side of the result, proud of the way the boys fought back," Smith said.
Opener Finch, who scored a quickfire 56, gave Australia a brisk start despite the early wicket of David Warner, who was out for eight after a scratchy 17-ball stay at the crease.
Finch took on the Sri Lankan bowlers with ease as he regularly cleared the infield to notch up seven boundaries and two sixes during his 46-ball knock.
Finch, who registered his 12th ODI half-century, fell to debutant left-arm spinner Amila Aponso after edging the ball to Sri Lankan skipper Angelo Mathews at first slip.
Mathews though could not hold on to a sharp chance at slip, giving Smith a new life at 15 off Dilruwan Perera and the batsman made the most of it.
Another debutant spinner Lakshan Sandakan got his maiden ODI wicket in Matthew Wade, who had worked hard for his 26 which came off 34 deliveries to put Australia in trouble at 128/3.
But Smith, who top-scored with a dogged 58, put together a 62-run partnership with George Bailey (39) to thwart Sri Lanka's spin attack on a sluggish pitch which was taking vicious turn.
Earlier half-centuries from Kusal Mendis and Dinesh
Chandimal helped Sri Lanka post 227 for eight in their 50 overs.
"I think the wicket played as it was right throughout, it only got worse. We were 20-30 runs short," said Mathews.
"We don't need to panic, just do the basics right, improve with bat, ball and field," he added.
Starc, who took three wickets, struck with the fourth ball of the innings to have opener Kusal Perera trudging back to the pavilion for one.
Dilshan fell for 22 but Mendis, whose 176 in the first Test remained the defining knock of the three-match series, registered his fourth half-century in only his eight ODI.
The 21-year-old got good support from wicketkeeper-batsman Chandimal, who scored an unbeaten 80, as the pair put together a 79-run third-wicket stand to help Sri Lanka rebuild.
But a double strike by Faulkner in the 30th over checked Sri Lanka's surge after Mendis and Mathews were dismissed in the space of three deliveries.
The left-armer broke a 19-year-old record set by Pakistan off-spinner Saqlain Mushtaq, who claimed 100 wickets in 53 matches.
The in-form Chandimal kept his focus to register his fifth consecutive ODI half-century after scoring four on the England tour.
But Faulkner took two more wickets to check Sri Lanka's run-rate in the final overs for a career-best ODI figure of 4-38.
The second match is scheduled for August 24 at the same venue.
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