Published 26/07/2015 at 11:28 GMT | Updated 26/07/2015 at 15:39 GMT
Hamilton's luck was awful from the off, making a terrible start alongside his team-mate, allowing the Ferraris of Vettel and Kimi Raikkonen to take first and second places in the first lap.
The defending champion then veered off the track into the gravel, blaming Rosberg for forcing him off, and then he was playing catch-up for the rest of the afternoon.
"Today was weird. Do I deserve any points? I didn't give up and drove as hard as I could," said Hamilton. "To come away from one of the worst performances I've put in in a long, long time... it shows we are human.
"I was all over the place. I don't have any words to explain what happened today. It was a really bad performance from myself.
"I pushed and never gave up but when I had two different choices I chose the wrong one every time."
Ferrari Formula One driver Sebastian Vettel of Germany (R) overtakes Mercedes Formula One driver Nico Rosberg of Germany (L) and Mercedes Formula One driver Lewis Hamilton of Britain after the start of the Hungarian F1 Grand Prix at the Hungaroring circui
Image credit: Reuters
A very unusual safety car procession followed about two-thirds of the way through the race, as Nico Hulkenberg's front wing collapsed, leaving all sorts of debris on the track and forcing the cars through the pit lane to allow the marshals to clear the carbon fibre obstacles.
Raikkonen suffered power problems in the latter stages, and Red Bull's Daniel Ricciardo drove brilliantly to cut through the field, eventually settling into third behind Vettel and Rosberg.
But when Ricciardo looked to dart past the Mercedes, there was contact, damaging the Red Bull's wing and ripping Rosberg's tyre to shreds.
Red Bull Formula One driver Daniel Ricciardo of Australia celebrates his third place after the Hungarian F1 Grand Prix at the Hungaroring circuit, near Budapest, Hungary July 26, 2015
Image credit: Reuters
It left the path clear for Vettel to cruise through the final two laps, and allowed Daniil Kvyat to take second. Ricciardo still managed a distant third - and Hamilton eventually clawed his way into sixth.