Rosberg won by 10 seconds from his teammate and two-time defending world champion Lewis Hamilton with Sebastian Vettel claiming the final podium place for Ferrari.
Former four-time champion Vettel had led until his final pit stop, having made a superb start off the grid to take early command at Melbourne's Albert Park.
It was Rosberg's 15th career victory and fourth in a row after dominating the closing races of the 2015 season, with Hamilton having already clinched his third title.
"It's early days but a perfect start. We need to keep our eyes on the red guys (Ferrari) and keep working," Germany's Rosberg told his team's official Twitter account.
Aside from the competitive nature of the tussle at the front, the major talking point was the spectacular crash involving former two-time champion Alonso.
His McLaren clipped the rear of the Haas of Mexico's Esteban Gutierrez on Turn 3 and flipped into the gravel, finishing hard up to the barrier.
To the refief of all concerned, the Spaniard was quickly extricated from his car and he was able to walk to the medical center to be checked out.
"I'm amazed that he walked away but it proves the safety of the cars now, Alonso's relieved teammate Jenson Button told Sky Sports, while Alonso himself also acknowledged he had been fortunate.
"I think it was a racing thing and sometimes we forget we are going 300 kilometers an hour," he added.
"It was a very, very scary moment," Gutierrez said.
"The first thing I saw in the mirror I was really worried. I jumped out as quick as I could and ran to him. It was a really big relief he was OK."
The race was halted after 18 of 57 laps under the red flag while the debris was cleared away.
But after the crash, the Mercedes pair were able to run to the finish without making a stop, while Vettel had to come in, leaving Rosberg to race to the finish unchallenged.
Hamilton, so impressive in qualifying first Saturday, had made an appalling start and immediately slipped to sixth place as Vettel and his Ferrari teammate Kimi Raikkonen charged to the front.
Raikkonen was forced to retire with a blown engine on lap 23, leaving the way clear for home hope Daniel Ricciardo in his Red Bull to take a popular fourth place from Felipe Massa in the Williams.
Romain Grosjean finished a superb sixth for American team Haas, the first time new entrants to Formula One had scored points in their opening race since Toyota back in 2002.
Nico Hulkenberg for Force India, Valtteri Bottas in the second Williams and the young Toro Rosso pair of Carlos Sainz Jr. and Max Verstappen completed the points scoring.
Dutch teenager Verstappen was frustrated though by not being allowed to pass his teammate and vented his spleen over race radio in strong terms.
New qualifying format 'scrapped'
Before the start Sunday, F1 chiefs announced they would be scrapping the new style elimination qualifying format, which was only introduced for the first time in Australia.
Designed to make qualifying more exciting, it appeared to have the opposite affect, with no teams on track in the final four minutes of the final session, Hamilton having already set the fastest time.
Following a meeting of team principals, it was decided to revert to the old format for the next race in Bahrain in a fortnight's time, but the change still needs to be ratified by the World Motor Sports Council.
"It shows there is common sense in F1," Mercedes F1 boss Toto Wolff said.