Charles Leclerc produced a stunning lap to pip Max Verstappen by 0.188 seconds and take his third pole in a row in Baku.
Hamilton had to settle for fifth on the grid, while team-mate George Russell will start down in 11th after failing to make Q3.
"This is not the position we want as a team," said Hamilton. "We exist to win. We haven't had an upgrade this weekend.
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