Pollard struck four sixes in 61 deliveries as West Indies reached its Duckworth/Lewis target of 158 with five wickets down and 10 balls to spare at Arnos Vale Sports Complex to level the five-match seriesat 1-1.
Allrounder Dwayne Bravo made 30 and shared an important fifth-wicket stand of 64 with Pollard.
It was West Indies' first ODI victory over Australia since October 2006, a stretch that included 13 losses and one no-result.
Earlier, Narine starred as Australia was restricted to 154-9 in a rain-shortened innings of 40 overs after being sent in to bat on a two-paced pitch. Fast bowler Kemar Roach supported him well by taking 2-23 as Australia struggled in the overcast conditions.
David Hussey top-scored for the visitors with 37 off 62 deliveries while there were useful contributions from captain Shane Watson (25), Michael Hussey (24) and George Bailey (21).
Australia was set back early, first by a lengthy rain delay that halted play at 16-0 from five overs, and then by two quick wickets upon resumption of play.
David Warner recived a ball that kept low from Roach to rattle his off stump. In the same over, Peter Forrest fell for a duck, edging a tentative stroke to second slip.
Watson hit a four and a six to break the shackles briefly but Sammy claimed him with a slower ball to make the score 46-3.
David Hussey shared restorative stands with brother Michael and Bailey to buoy Australia.
The Husseys put on 31 for the fourth wicket while David Hussey and Bailey added 32.
Michael Hussey, dropped by Pollard on 8, eventually provided Narine with his first scalp, edging to wicketkeeper Carlton Baugh.
Bailey fell to Dwayne Bravo to make it 109-5, slapping the ball to backward point.
The lower order failed to contribute much before being mesmerized by Narine.
Wade top-edged to short fine leg, David Hussey was eventually bowled off the inside edge, while Clint McKay was stumped as the innings fizzled out at the end.
Watched by a sell-out crowd of 11,000 that included out-of-favour former captain Chris Gayle, West Indies began disastrously when Kieran Powell was lbw to Brett Lee's first ball.
But Johnson Charles and Marlon Samuels revived the innings with a stand of 42 in 10 overs.
Australia made its way back into the contest with three cheap wickets in the middle overs. Samuels was bowled by a yorker by Watson, while Charles was run out after a mix-up with Darren Bravo, who fell soon afterward when he dragged on from Xavier Doherty.
The innings was at the crossroads at 74-4, but Pollard and Dwayne Bravo steered the hosts to victory.
Pollard was circumspect but opened his shoulders when the opportunity presented itself and launched four sixes, including one that sailed over the Michael Findlay Stand at deep midwicket off Doherty.
The run out of Dwayne Bravo with 20 still needed off 40 balls gave Australia a sniff. But Baugh lashed a four and a massive six to finish the match.
The third contest in the five-match series is at the same venue on Tuesday, and Primer Minister Ralph Gonsavles ensured another big crowd by on Sunday declaring that day a public holiday.