The 21-year-old's performance at Albert Park made him the first Dane to stand on an F1 podium, as well as the highest placed debutant since Canadian Jacques Villeneuve made a sensational start with Williams in 1996.
The Woking-based team had been off the podium since 2012 but Magnussen, son of former F1 racer Jan, led the way back with team mate Jenson Button joining him in third after Red Bull's Daniel Ricciardo was disqualified from second in his home race.
"It does feel incredible. It's now crazy to think that we are here," Magnussen told reporters before Ricciardo's disqualification, which Red Bull have appealed.
"Today I've done my first Formula One race, I got on the podium and I did it with McLaren.
"It's such a dream come true. I can't explain how it feels, it's just unreal."
The Dane was fast-tracked into Formula One by McLaren after he won the Renault 3.5 World Series title last year and impressed in a young driver test.
Lewis Hamilton, in 2007, was the last rookie to make a Formula One debut for McLaren and he finished third in Melbourne that year and challenged for the title all season before winning it in 2008.
McLaren felt Magnussen was so promising they dropped Mexican Sergio Perez to make way for him.
The Dane, whose father's F1 career stalled and ended with just one point after he made his McLaren debut in 1995, said it was a "gamble" that had paid off.
"To take a rookie is a gamble in Formula One, especially in a top team," he told Sky television. "They listened and believed in me and I can't thank them enough. This has been the best day of my life but I'm sure I'll have better days ahead."
Though Hamilton and reigning world champion Sebastian Vettel were sidelined by reliability problems, Magnussen put two-time F1 champion Fernando Alonso (fourth) and his Ferrari team mate Kimi Raikkonen (seventh) in the shade.
"Despite his youth and inexperience, he drove like a man who'd notched up 100 grands prix already," said McLaren Racing Director Eric Boullier.
The season-opener in Melbourne had promised to be a lottery as teams grappled with the sport's new technological revolution which includes V6 turbocharged hybrid engines instead of the screeching old V8s.
While Ricciardo's disqualification left a sour note for Red Bull and local fans, the 24-year-old's composure over his first race weekend with the team will not be forgotten.
The Australian had qualified second in a rain-hit session on Saturday and Sunday was another boost for the champions after a nightmare in pre-season testing.
Vettel's record nine-race winning streak was broken when he retired early on but he stayed on to see another of his milestone achievements disappear.
Toro Rosso's Russian rookie Daniil Kvyat finished ninth to become Formula One's youngest points scorer at the age of 19, eclipsing the previous record set by Vettel at the 2007 U.S. Grand Prix.
"It's a great feeling. It was quite a good race for us, we managed to keep the rhythm very high," the confident Russian said.
"I think we did a really good job. We were quite good on tyres. It feels like a good step, every session we are learning something."
Toro Ross principal Franz Tost was sure there was much more to come.
"I'm very happy for Daniil...This is very promising and I'm convinced that this is the first of a long series," he said.
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