Norris and team-mate Oscar Piastri finished second and third at Suzuka as McLaren scored its first double podium of the season.
McLaren started the season with a pair of scoreless races but has since developed the MCL60 into what is more often than not the second-best car on the grid, behind only Red Bull's constructors' championship-winning RB19.
Although Norris finished 19 seconds off Verstappen in the race, team-mate Piastri was comfortably clear of fourth-placed Charles Leclerc at the chequered flag as he scored his first F1 podium.
"Another amazing day for us," said Norris. "P2 and P3, so we couldn't have asked for any more.
"The team did an amazing job. My start was very good. I almost had Max, but Max is Max as well. I didn't have a lot of chance in Turn 2, but I tried.
"The pace was extremely strong today, compared to everyone. And I mean, we're not close to Max, but we're not miles away either. So it was a very good day. I'm very happy also for Oscar, his first podium in Formula 1. So congrats to him.
Pressed on the margin to Verstappen and Red Bull, Norris added: "We're pushing, we're getting there.
"The progress we've made is pretty outstanding. Like I said, I'm very proud of the team, and the steps forward we're making every weekend.
"I'm sure there's gonna be some tough times to come. But we're getting there step by step. And our first double podium together with Oscar. So yeah, a good moment for us."
On just his 16th race weekend, Piastri secured his first F1 silverware on a Sunday with a confident drive after initially dropping from his front-row berth into third when attacking Verstappen off the line.
"It feels pretty special. Definitely, I will remember for a very, very long time, so I can't thank the team enough for giving me this opportunity," said Piastri.
"There are not many people in the world that get this opportunity in their whole life. And I've managed to have it in my first season. So thank you very much to the team.
"It wasn't my best race ever. But it was enough to get a trophy at the end. So yeah, super happy."
