Norris headed the Q2 times on medium tyres, as F1 rolled out a new 'alternative tyre allocation' trial that stipulated which compounds teams must run in each qualifying segment, but came up just short in Q3 following the switch to the soft rubber.
The Briton was denied a second career F1 pole as he clocked a 1m16.694s that was just 0.085s slower than pacesetter Lewis Hamilton's Mercedes, who pipped Max Verstappen's Red Bull by 0.003s to end a pole drought dating back to Jeddah 2021.
Speaking after the session, Norris acknowledged that he was "a little bit" surprised to show so well in Hungary, before conceding his frustration.
"I'm disappointed, I guess," he said. "If you're within a tenth of pole, it feels like you should be on pole if you put the lap together."
On a strong day for McLaren, following on from the British Grand Prix last time out where Norris and Oscar Piastri started second and third, the papaya team locked out the second row with Piastri lining up fourth.
But Norris admitted to making "too many mistakes" to be totally satisfied with his performance.
He added: "I guess, as an overall picture, I'm happy the team did a good job, P3 and P4 for both of us. So a good weekend so far.
"But as a driver, I'm not the happiest. I feel like I made too many mistakes and that cost me today."
He explained that his final Q3 lap "wasn't the cleanest" as he tried to weigh up the balance of risk versus reward, and conceded there was "definitely a tenth in it" to be found.
"There's always an element of risk involved in trying to push a little bit more," Norris said.
"P3 is still a good position for tomorrow, so still a good day."
Norris believes McLaren's race pace on Friday "was pretty strong, just as strong as we probably were in Silverstone" and feels "excited" for the team's prospects in the race.
But he cautioned that the Hungaroring's paucity of overtaking opportunities could make life hard in the race and was "probably why I'm a bit more frustrated than I normally would be".
"I'm up there with the good guys, we've got two cars up there," he went on.
"So hopefully we can use them and get some good points tomorrow."
