Lewis Hamilton suffered a shock exit from Q2 in a dramatic qualifying session that also saw his teammate Charles Leclerc crash out.
The extended qualifying session on Saturday was red-flagged six times, with Leclerc, Pierre Gasly, Franco Colapinto and championship leader Oscar Piastri hitting the barriers.
Seven-time world champion Hamilton will start 12th on the grid after his best Q2 lap put him 10th quickest, before Leclerc's faster lap time dropped him into the knockout zone.
Leclerc progressed into the final session of qualifying before he crashed into the barriers at Turn 15 in his first attempt at a pole position shootout with seven minutes remaining.
"I'm obviously so disappointed," Hamilton told Sky Sports. "Yesterday the car was feeling good, today there was a direction that we ended up going, which on paper looked like it was the best place for us to be.
"Our pace had been good. We'd been progressing, I was feeling really on it, didn't make any mistakes, didn't see me down any exit roads. It was just that we didn't have the right tire on at the end.
"It's tough. Everyone ahead of me basically had the medium tire on, but I lost a medium tire in P2 due to run-plan schedule, and that put me on the back foot."
Hamilton, who will start behind Fernando Alonso, added: "As I said, there's been lots of positives from this weekend. I've really felt on it. I honestly thought I was going to be shooting for pole today, so it's kind of a bit of a shock.
"I'll take it on the chin and keep trying."
Leclerc, meanwhile, said he never quite felt comfortable with the car at a track he usually excels at.
"I've been struggling massively, and I come to Baku, and everything flows pretty naturally, and this year it wasn't the case," he said. "I don't think I understand why.
"In qualifying, I changed a lot of the car to try and change the feeling and the feeling was getting better until we actually put [on] the medium tires and everything went ... yeah.
"It was very, very difficult from the moment we put the medium on. For some reason yesterday it felt like the better tire on our car. Today it felt like we could never could reach the right window of the tires and that made it very tricky."