The Briton finished 11th in FP1 on Friday, which he says was "probably one of our worst FP1s of the season in terms of pace," before going on to finish 14th in FP2.
He improved to 10th in a rain-hit FP3 on Saturday morning before finishing third in qualifying, despite saying he "wouldn't say we felt anywhere near this good."
Asked whether he was surprised by the result, Norris said: "I would say yeah. After FP1 we said it was probably one of our worst FP1s of the season in terms of pace. So to be sat here today, I think it's quite a big surprise for all of us.
"I think the cooler conditions, things just started to come our way a little bit.
"So yeah, it just feels a bit odd, I definitely wasn't thinking at all to be in the top three today, but I'm very happy to be here, so a good day."
He added: "I'm shocked to be here. We've not really brought anything to the car this weekend, the last upgrade we had was Baku with the floor and even that didn't give us a lot of performance, just a different direction to go in.
"We've not really brought too many big things, just some very small things that obviously help a little bit.
"In a good way, I'm happy, but we didn't really change anything this weekend, it's the same car that we've been struggling with, so managed to see some good points from it."
Norris said he thought it was "going to be tough to get out of Q1" after Friday's running, which saw team-mate Oscar Piastri also finish outside the top 10.
He accepted that the car is "still pretty poor in many areas" but acknowledged that the new layout, with the chicane removed, has helped the team.
Norris was the first driver onto the intermediate tyres in FP3 before also being the first to switch to slicks, and says he tends to test different strategies "for my own feeling and confidence."
Asked by Motorsport.com whether his FP3 strategy had given him a confidence boost into qualifying, Norris said: "I feel like I tend to always be one of the first to try these things for my own feeling and confidence, I tend to do quite a bit.
"I felt decent, I wouldn't say we felt anywhere near this good, and like I said on Friday, we all sat down and were thinking it was going to be tough to get out of Q1.
"I know sometimes we outperform that quite a bit and we do well, but I think apart from maybe Bahrain this was our worst Friday of the season.
"I really wasn't that hopeful but things just seemed to come together, I felt like other people struggled a bit more today than maybe us overperforming or being our true potential.
"Like I say, we've not really changed the car that much, so there was nothing which was giving me a lot of confidence or any reasons we were going to be that good today.
"We're still pretty poor in many areas but high speed has always been one of our strengths for years and slow speed quite the opposite so I think this layout has helped us, certainly benefitted us a bit more.
"I think already FP2 I was flat in Turn 3, flat in Turn 9, probably one of the few guys apart from Red Bull who did that.
"It's a track where more of the strengths are able to be shown. Even though Monaco wasn't bad, we do struggle in the low-speed corners, so that's the area we know we want to take big steps forward."
Additional reporting by Alex Kalinauckas
