The Briton, who started the season-opener on pole, took control in Melbourne mid-way through the race after a wheel-to-wheel battle with Ferrari's Charles Leclerc in the early stages.
He came home ahead of teammate Kimi Antonelli by nearly three seconds and had a more than 15-second gap to Leclerc to reinforce his status as the pre-season favourite.
"Honestly, thank you so much to the whole team because it's been a long time coming to have this car beneath us and we couldn't start off in a better way," he said.
But Russell is not getting carried away with the team still getting on top of new regulations around the engines and chassis and a new challenge next weekend in China with a sprint race.
"It just feels like another race win, to be honest. I mean, it's race one into a very long season," he said.
"Of course I want to fight for race wins week in, week out, but we're all here now to fight for a world championship, and that's what we've been working so hard towards.
"And if we want to do that, we still need to raise our game because there were a lot of areas today that we underachieved, mainly around the race start and having the battery in the right place."
Russell dominated qualifying and started at the front of the grid, but he was overtaken by Leclerc at Turn 1, which set off a thrilling battle between the pair with their pit-stop strategies proving decisive.
"Qualifying was a real surprise for us and I think we saw today that Ferrari are definitely there in the mix," he said.
Antonelli had an even worse start than Russell, sliding to seventh from second.
But the teenager clawed back to secure a well-deserved podium after a testing weekend that involved a big crash in third practice.
"The start was poor, very poor, and I just lost a lot of places and found myself having to chase," said the Italian.
"But the car was very strong and it was good fun at the end. Team did an incredible job."
