Smith's third ODI ton was instrumental in helping the home side chase down a steep target of 304 with one ball to spare after the middle-order failed to offer him steady support.
For England, Ian Bell's 141-run knock went in vain as Smith stole the thunder, continuing his blistering batting form this season. England would now hope for India to lose Monday's match to Australia to make the finals.
Smith's knock came off 95 deliveries and included half a dozen fours and a towering six.
Finch, who scored at run a ball and struck three fours was yorked by off-spinner Moeen Ali, which was followed by a couple of quick dismissals.
From a comfortable, 253/3, the Aussies were reduced to a shaky 275/5 before Smith and Haddin steadied the ship.
However, next man in Smith continued his stupendous run of form even as he struggled to get support at the other end. While Cameron White departed without disturbing the scoreboard, Glenn Maxwell and James Faulkner failed to convert their fine starts.
Maxwell shared a 69-run fifth-wicket stand with Smith.
Faulkner, on the other hand, fell after adding 35 runs to the total and sharing a 55-run partnership with Smith for the sixth wicket. The all-rounder was dismissed by pacer Chris Woakes.
Smith finally found a firm partner in vice-captain Brad Haddin, who went full throttle in the last 10 overs to take the pressure off his skipper. Haddin was eventually sent back following a 29-ball 42, during which he smashed four fours and a six.
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