Both the Mercedes drivers and Vettel were putting in solid times as they sought to narrow the gap to Leclerc before his stop.
"I honestly did not think we would be on the front row", Hamilton said to Sky Sports.
"I think we gained relative performance with the dropping temperatures", Wolff added.
37/51 - After pitting for 2.8 seconds, Leclerc uses DRS to overtake Gasly and return to fifth.
"I just didn't get the job done in qualifying".
"Some people like to put it that this is the "turning around" or "the decider" or so on", he said of the race through the streets of Baku.
7/51 - Max Verstappen has had a sloppy start and is three seconds behind Vettel in fourth. We'll have to wait and see if that chassis is repairable or if we have to go to our third chassis.
Why, Mercedes felt, give their drivers the complication of having to adapt from soft tyres to medium and then back again in the middle of qualifying when things were already hard enough anyway? The Mercedes pair of Bottas and Hamilton are now battling each other for second and the chance to take on Leclerc. "It's up to them to go and check them all, there's plenty of people here, but the same thing has happened before here".
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The reality is it's a trial and error process to find the right combination - and while she was dealing with that she also had to cope with her dad's recent illness .
He added: "Two wins in four races is a lot better for me than a year ago so I am feeling good about that".
"We need to be faster".
Bottas started from pole position and kept Hamilton at bay as the Mercedes duo went wheel-to-wheel through the opening three bends.
Bottas was on course for victory in Baku previous year when a blown tyre three laps from the end handed it to Hamilton instead.
Hamilton attempted a late charge for victory, but Bottas had enough in his pocket to take the chequered flag and move one point clear of the Brit in the title standings.
He had been 0.3secs to the good after the first runs, but those errors cost him that amount of time at the start of the decisive lap.
Vettel is now in third place in the driver's championship, 34 points adrift of Lewis Hamilton in second. "I'm better, still not 100 per cent but going in the right direction".
"All weekend, we haven't really been on it, and that's why getting into the points would have been a fairly good recovery for us, that's why I'm obviously gutted not to pull that off".