The 21-year-old Dutchman's best lap in a Red Bull was clocked at one minute and 14.572 seconds.
The Mercedes duo of Valtteri Bottas and Lewis Hamilton will start second and third respectively at the Hungaroring.
"I am very happy," said Verstappen of his achievement in his 94th Grand Prix. "There's still a race to do and at the end of the day, that is the most important but it's very nice to get a pole position and it is great for all the team."
Verstappen has won two of the last three Grand Prix to propel himself into third place in the drivers' championship with 162 points. Bottas is second on 184 while Hamilton boasts 225 points after 11 of the 21 races in the 2019 season.