World champion Verstappen held on to provisional pole after the first of two hotlaps with a 1m40.445s in his Red Bull RB19, with Leclerc posting an identical lap time to go second, because the Dutchman had set the time first.
On the deciding run, Verstappen could only find a minimal improvement to a 1m40.391s, while his Ferrari rival put two tenths on him with a 1m40.203s.
It's the first time this season that Red Bull has failed to grab pole, with Ferrari having the edge on the world champions through Baku's tight corners.
Afterwards, Verstappen explained a different approach on his out-lap may have cost him time.
"It's always tough around here to really put the whole lap together," he said. "I think also the second run we tried something different on the out-lap, which maybe was not ideal for the lap time at the end.
"But around here, it's just really hard. In Q3, you're on the limit and then trying to make everything perfect; it's not the easiest.
"But nevertheless, we're P2 we know that we have a very good race car. So, all in all, it's not bad, but you always want to start ahead, but we'll have to pass one car."
Source: Autosport