The Monegasque started on pole position in Baku, and will do so again on Sunday's main Azerbaijan Grand Prix, but was unable to prevent Mexican Sergio Perez from winning and taking the full eight points.
Red Bull's double world champion and series leader Max Verstappen was third, beaten by someone other than his teammate for the first time this year.
Leclerc's seven-point haul was still more than he had taken from the first three rounds of the season and gave him hope for the future.
"It confirms a little bit what we thought: the Red Bull still has the upper hand in the race. But we, again, must not forget how far we were behind on race pace two races ago," he told reporters.
"So we did a step forward; we are still not where we want to be. But...if winning is not possible then we just need to take the maximum points. And today, there wasn't anything more, so (I'm) happy with the second place.
"And we will try, anyway, to go for the win tomorrow, even though the Red Bull seems to be quicker."
Ferrari failed to win any points in the previous race in Australia, with a five-second penalty demoting Carlos Sainz demoted from fourth to 12th, but they were adamant then that progress was being made.
Team boss Fred Vasseur said Saturday had provided more evidence of that.
"In Melbourne, we saw we had made progress and I think that so far this weekend, that has also been confirmed here," said the Frenchman.
"In qualifying trim, we were quicker than Red Bull and in the race, we were able to fight with Perez and Verstappen, even if we saw there is still some work to do in terms of pace."
