"Yes, pole position, numero uno," their Ferrari team shouted over race radio as the German driver clinched the 48th pole of his career.
"It was seamless, we didn't have any problems," Vettel said. "The car was fantastic today. It was really a pleasure to just go round."
Vettel is optimistic he can secure a fourth win of the season and expand his advantage over Lewis Hamilton heading into the summer break. He leads his title rival by just one solitary point after 10 races.
"Chances are good. The balance of the car was better today. I don't see why it should change tomorrow," he said. "But it's always hot here, tough on tyres."
The four-time F1 champion clocked a best lap of 1 minute, 16.276 seconds, while Raikkonen was .168 behind in front of a large contingent of flag-waving Finnish fans.
Vettel stood on top of his Ferrari and waved to the crowd. He had extra reason to be happy, with both Mercedes cars behind him on a track where overtaking is notoriously difficult.
Valtteri Bottas was third, .254 behind, while Hamilton could only manage fourth, .431 behind, as the British driver missed out on equaling Michael Schumacher's all-time pole record of 68. Hamilton won the British GP two weeks ago while Vettel could manage only seventh.
"Obviously, the last race wasn't great for us, but it doesn't matter now," Vettel said. "The main task comes tomorrow."
This time, it appears that Mercedes is on the back foot.
"They clearly got the upper hand today. We clearly have a lot of work to do," Bottas said. "Clearly, we are not performing as we want."
Along with Monaco, the tight and twisting Hungaroring is the most difficult circuit to overtake on, making the task facing Mercedes even harder.
"For sure the start will be important," said Bottas, who has won two races and is in third place overall.
Max Verstappen will start from fifth - ahead of his Red Bull teammate Daniel Ricciardo, whose car was repaired in time after breaking down during the third and final practice session on Saturday morning.
Fernando Alonso qualified in a promising eighth for McLaren, who have been struggling with Honda engines this season.
Meanwhile, British driver Paul di Resta climbed into an F1 car for the first time since November 2013. He was an emergency late replacement for Felipe Massa, who pulled out after complaining of "dizzy" spells following Friday's practice.
Di Resta did well given the circumstances, placing 19th and relegating Sauber driver Marcus Ericsson to last. Pascal Wehrlein (Sauber), Lance Stroll (Williams) and Kevin Magnussen (Haas) were the other drivers eliminated from Q1, the first part of qualifying.
Vettel had the fastest time in Q1, while Hamilton complained of vibration on his car. But with his father, Anthony, watching on, Hamilton shot to the top of the leaderboard right at the end of Q2, raising hopes of a pole.
The five drivers eliminated from Q2 were Romain Grosjean (Haas), Sergio Perez and Esteban Ocon (Force India), Daniil Kvyat (Toro Rosso) and Jolyon Palmer (Renault).
Hamilton led early into Q3, only for Vettel to go even quicker and draw a roar from the crowd taking in the warm sunshine around the 4.4km circuit nestled in the hills surrounding Budapest.
Earlier, Vettel set the fastest time in third practice, nearly half a second clear of Raikkonen. Hamilton could only manage the fifth-best time: 1.4 seconds behind Vettel. It proved an indicator of what followed in qualifying, and what might lay ahead in Sunday's race.