Raikkonen was the only driver to lap in the 1:29 bracket.
Hamilton was serving a five-place penalty for changing his gear box so early in the season and he fought from ninth to earn a third-place finish.
There would be light contact between the Mercedes and Red Bull, however, with the Dutchman picking up a puncture, at the same time up ahead, his teammate Daniel Ricciardo was pulling over to the side of the track after his vehicle shutdown.
"I've got the softer tyre so I should be able to go longer [before pitting]".
Verstappen had already set a time fast enough to get into Q2, but the vehicle was too damaged to take any further part in the session. Fourth position for Pierre Gasly was helped by both Red Bulls and Kimi Raikkonen failing to make the finish, but the fact he finished comfortably clear of a Haas, Renault and McLaren was not something everyone was expecting over race distance.
Sixth is the highest the four-times champion can start Sunday's race - which has never been won from lower than fourth on the grid - even if he sets the quickest time in qualifying. There were also celebrations at Sauber, with Sweden's Marcus Ericsson finishing ninth for his first points since 2015 and the Swiss team's first of the season.
'I go when the light is green and I don't see what happens behind.
Ronaldo scores again but Madrid derby ends all square
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Israeli military finds two explosive devices planted on Gaza border
While political leaders brushed aside complaints , on Sunday the IDF said it would investigate its own conduct in the confrontations.
Turnout high in Hungary's election as Orban fights to retain power
After casting his vote in a wealthy district of Budapest with his wife, Orban said Hungary's future was at stake in the vote. Such an outcome would allow Orbán to pursue radical change, perhaps tightening his grip Hungary in the same vein as Poland.
"Their engine mode is easily as good as ours". Ricciardo coasted to a halt at Turn 8, bringing out the Virtual Safety Car, with the incident compounding a miserable few moments for Red Bull as Max Verstappen was limping to the pits with a puncture.
Now he needs to start winning again - starting in Shanghai. "Better too with their tyres". The Mercedes seems to be more comfortable on this compound in comparison to the supersofts, evidence that tyre overheating could be an issue of the W09.
Brendon Hartley fell agonisingly short of reaching Q3 and will start from a still impressive P11. Raikkonen in third was 16s ahead of Hamilton but recorded his second DNF of this season when he was forced to stop in the pits due as Ferrari failed to change his rear left tyre. He was the highest placed driver outside of Ferrari, Red Bull and Mercedes.
Saturday's pole was the 51st of Vettel's career and third in Bahrain.
"Both Mercedes at the end of the stints were very strong", said Vettel. I was happy with my first run until I tossed it away in the last corner so I stayed away from that kerb in the second run. "There are a couple of different strategies tomorrow so I will try and eek out every last bit of power and strength from the auto". "The auto was excellent all weekend so far".