Leclerc clocked a best lap round the Albert Park circuit of one minute 17.277 seconds, 0.381 clear of Red Bull's Verstappen, who won the first two races of the season.
"It was a solid first day, everything felt good from the start and I was comfortable in the car," said Monaco's Leclerc, who was third in Saudi Arabia a fortnight ago.
"It looks good for now but we have to wait and see how it will look tomorrow. I expect the field to be much tighter."
Verstappen ran over a kerb late in first practice in Melbourne and damaged the floor of his car, and the team were still working on the issue when the second session started.
The Dutchman eventually emerged as the last driver out after missing 22 minutes and he struggled initially to make inroads on medium tyres as others switched to softs.
But when he made the tyre change he began flying to narrowly miss topping the timesheets at a circuit where he triumphed from pole last year.
"Today was a little bit messy because of what happened in FP1. We had some damage to the floor and chassis, which took a bit longer to fix, meaning I had a bit of catching up to do," he said.
"I think we know what we have to focus on and we have a few things to fine-tune, but we are looking forward to tomorrow."
The Red Bull ace won in Bahrain and Saudi Arabia to start the season, but those victories were overshadowed by turmoil within the team camp.
The Dutchman is hoping to put the off-track distractions behind him with records on the line in Melbourne.
Verstappen is chasing a 10th consecutive victory to match his own record feat last year, while Red Bull are attempting to score a third straight 1-2 finish for the first time in the team's history.
Third in second practice was a remarkable achievement for Ferrari's Sainz, who underwent appendicitis-related surgery and missed the last race.
There was concern about how his body would fare, but he said he felt "reasonably comfortable".
"I will continue to take it step by step to make sure I come to qualifying and to the race in the best form possible," added the Spaniard.
The Aston Martins of Lance Stroll and Fernando Alonso were fourth and fifth ahead of George Russell.
Russell's Mercedes teammate, the seven-time world champion Lewis Hamilton, could only finish 18th, some 1.5secs behind Leclerc. He said on the team radio "something's wrong".
McLaren's Oscar Piastri was seventh ahead of Sergio Perez in the other Red Bull and Lando Norris.
RB driver Yuki Tsunoda rounded out the top 10, which was covered by less than one second.
Albon smash
Under blue skies, Valtteri Bottas and Perez led the way out of the pits, with the Red Bull driver setting the pace.
But he was quickly bettered by Alonso and a host of others.
The times kept coming down with Stroll on top at the halfway mark.
Leclerc then took control and Verstappen began ominously climbing the leaderboard when he switched to softs, but the Dutchman was unable to better his rival.
In an eventful first practice, Williams' Alex Albon smashed into the wall at turn 8.
It caused heavy damage on the right side of his car and left debris littering the track. Albon was uninjured but did not compete in FP2.
There were several other mishaps in a reminder of the dangers of a track that witnessed multiple crashes last year, with just 12 drivers completing the race.
