The Briton, who is 14 points behind Formula One championship-leading team mate Nico Rosberg after 10 races, had been fastest in all three practice sessions and was favourite for pole position.
Instead, the 2008 world champion, who has won for the past two years in Hungary and four times in all, faces another fight through the field in Sunday's race at a slow circuit where overtaking is difficult.
"Guys, I'm on fire," Hamilton, who has had a run of bad luck this season with two race retirements and qualifying problems as well as five wins, said over the car radio before coasting to a halt.
"OK Lewis just stop the car where there is a fire marshal," his race engineer replied. "I can't stop it," said the Briton, who finally did so at the pitlane entry and scrambled out as marshals extinguished the blaze.
He then walked away, helmet still on and head down. Hamilton slapped the barrier lightly in frustration as he went.
"There's a lot going through my mind, but I just have to try to turn it into positives until tomorrow," Hamilton told the BBC. "I think it's getting to the point beyond bad luck - it's something else. We just need to do better."
Mercedes said a fuel leak, the source of which was unclear, had led to the fire in the engine.
The 29-year-old had to start last weekend's German Grand Prix in 20th place after he crashed in the first phase of qualifying due to a front brake disc failure. He ended up finishing third.
Hamilton said the team would have to change the engine and gearbox, bringing with it a mandatory five-place penalty. That will mean he is sure to start in the pitlane.
"I honestly don’t know what I can do tomorrow," he told Sky Sports television.
"This is a track that you cannot overtake on so I think I will struggle to get in the top 10 tomorrow or at least the top five. I will probably leave here more than 20 points behind Nico, but there are still races to go.
"I don’t know what to say, I will try my best."
(Reporting by Alan Baldwin, editing by Tony Goodson)
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