The Spaniard qualified a distant 19th, with team mate Kimi Raikkonen 20th of 22 drivers, after the Italian team made a series of wrong calls on tyre selection on a wet and gusty afternoon.
Alonso will move up to 17th and Raikkonen 19th after penalties are applied to other drivers.
It was the first time since 2010 that Alonso had failed to make it through the first phase of qualifying that normally separates the tail-enders from the rest.
Both the Williams cars, who had started the previous race on the front row, also failed to make it through the opening session.
"We need to do better next time. I think the bigger teams have longer procedures than smaller teams, so we need to speed up some of the communications and some of the things that we do," Alonso told reporters.
"At complicated moments like this, you need to fit the right tyres at the right time, but luck definitely plays a part," he said.
Technical head Pat Fry said qualifying, after Alonso had been third fastest in Friday practice, was bitterly disappointing.
"It means we will have an uphill struggle tomorrow, with a plan to attack right from the first to the last lap," said the Briton.
He explained that conditions were mixed at the start of qualifying, with a lot of water on some parts of the circuit while others were drying or dry.
Williams, who had Finland's Valtteri Bottas and Brazilian Felipe Massa 17th and 18th in qualifying (although they will start 15th and 16th), were equally disappointed.
"It was down to seconds of timing and ourselves and Ferrari didn't get the best of that opportunity," said Williams engineering head Pat Symonds.
"We didn't see that rain coming, to be honest."
(Editing by Josh Reich)
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