Verstappen, the world champion, endured another frustrating race when a mechanical failure ended his hopes while he was running second 39 laps into the 58-lap race.
Verstappen, who was clearly disappointed after the race, never really challenged Leclerc, whose Ferrari was the quicker car under race conditions in Melbourne.
The drivers' championship leader, who pitted on Lap 22, confirmed that when posting the fastest lap of 1 minute, 20.260 seconds with a final-lap flourish.
He is the first Ferrari driver since Fernando Alonso in Singapore in 2010 to complete a racing "grand slam," having claimed the pole position, led wire-to-wire, claimed the race and also posted the fastest time.
"Honestly, what a car today. Of course, I did a good job all weekend, but it was not possible without the car," he said.
His teammate Carlos Sainz, who finished on the podium in Bahrain and Saudi Arabia, had less luck in a testing weekend.
After an issue in qualifying resulted in him starting from ninth, he dropped back further in the field with a slow start and then lost control on the second lap, ending his race.
But Leclerc believes Ferrari has demonstrated it is going to be competitive in the manufacturers' championship this year.
"Obviously we are only in the third race, so it is difficult to think about the championship," he said. "But, to be honest, we have a very strong car, a very reliable car too, and for now we have always been there. I hope it continues like this and if it does, we probably have chances for the championship."
The Mercedes team has endured problems so far this year and is off the pace of their Ferrari and Red Bull rivals.
But Russell was pleased with the improvement in his car this weekend, though he conceded they received some luck with Verstappen's misfortune.
"We got a little bit lucky today … but we will take it. To be standing on the podium is special," Russell said.
"We are never going to give up. We are going to keep on fighting. We have to keep this up while we are on the back foot, I'm sure we will get there after a few more races."
Verstappen said the Red Bull was "already miles behind" Ferrari.
"I don't even want to think about the championship at the moment," he said. "I think it is more important just to finish races, because today was, in general, just a bad day again. It is pretty frustrating and unacceptable.
"I knew there was a problem and it was always going to be a question mark for finishing the race, but these kind of things, if you want to fight for the title, cannot happen."
-AP
