Verstappen finished second while seven-time world champion Lewis Hamilton was third.
Mexican Sergio Perez, winner of the previous race in Monaco, was second on the streets of Baku.
Leclerc's car was dominant on the upgraded Albert Park circuit as the Monagesque sealed his fourth career win more than 20 seconds ahead of Red Bull's runner-up Sergio Perez.
The Monegasque, who started from pole, crossed the line 5.5 seconds ahead of Spaniard Sainz.
Velas Network will be creating create digital content for the Ferrari fans.
Team boss Mattia Binotto told reporters that the 22-year-old German would be available to stand in at 11 of the record 23 scheduled races should Charles Leclerc or Carlos Sainz be unavailable.
The Spaniard halted the session when he smashed head-on into the barriers at the Ascari chicane with just over half an hour remaining.
The family have kept updates to a minimum since the accident and in an upcoming Netflix documentary, set to premiere on September 15, Corinna said privacy was very important to them.
They are reserved for the house's selected and most faithful clients, those who already own one or more Ferraris.
"I wanted that MSC," the 21-year-old Haas driver told reporters during pre-season testing in Bahrain ahead of his F1 race debut there on March 28.