Reigning world champion Hamilton led from the start to the chequered flag, beating last year's winner Rosberg by 1.3 seconds, with Sebastien Vettel third in his Ferrari debut.
Only 11 cars finished the race after mechanical problems and an injury to Valtteri Bottas left just 15 on the starting grid -- the lowest number for a year-opener since 1963.
And Max Verstappen's record-breaking debut at the age of 17 years and 166 days ended on a sour note when his Toro Rosso suffered a power unit failure on the 34th lap.
It was Mercedes' eighth straight victory, dating back to Belgium last August, and their 20th race in a row with at least one driver on the podium.
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Hamilton, 30, has now won seven of the last eight races and 34 from 149 grands prix, although this was his first in Australia since 2008.
"Nico was quick throughout the race," he said of his fierce rival. "In terms of tyres we did not know how far they could go. When Nico turned up the heat I was able to react which was good."
"Lewis has done a fantastic job, he drove like a world champion all weekend. I'll give my maximum and try to beat him all year," he said.
Four-time world champion Vettel yelled "Forza Ferrari!" as he celebrated a morale-boosting podium finish on his debut for the team after switching from Red Bull.
Ferrari finished only fourth in last year's constructors' championship and Vettel, who beat Williams' Felipe Massa to take third place, was all smiles as he collected his trophy.