The three-time world champion, who leads Norris by 57 points in the drivers' championship with five races remaining and 146 points up for grabs, suffered from engine issues on Friday before bemoaning a lack of grip as he finished five tenths off the pace in final practice.
But Verstappen strung it together on his one timed lap in the final session of qualifying to claim second on the grid, less than a tenth clear of third-placed Norris.
The pair had no answer to a blistering lap from Ferrari's Carlos Sainz, with the second Scuderia of Charles Leclerc - last weekend's race winner - taking fourth.
George Russell claimed fifth on the grid for Mercedes, a place ahead of team-mate Lewis Hamilton.
Verstappen was delighted to put his Red Bull on the front row, especially after title challenger Norris had topped the first two sessions of qualifying.
"To be on the front row is an incredible result for us," Verstappen said.
"Yesterday was basically a complete write-off. We were behind and the car was not feeling great. Everything was very difficult.
"So it was always going to be a tough qualifying, we made some adjustments and it started to feel better.
"I don't expect miracles (on race pace). In Austin, Ferrari were really fast, McLaren were competitive, so from my side I guess we will find out tomorrow."
Mexico City's Autodromo Hermanos Rodriguez features the Formula One calendar's longest run down to the first braking zone - a 768-metre charge that could be critical when it comes to the title race.
Norris said on Thursday that he would "adapt" his driving after being penalised for overtaking Verstappen off the track in Austin last weekend, hinting he may need to be more aggressive in certain situations.
That could be put to the test on Sunday if he wants to get the better of Verstappen on the opening lap.
Norris believes third is a reasonable result and hopes his McLaren can deliver strong pace for Sunday's 71-lap race.
"I am relatively happy to be P3," he said. "I felt a bit on the back foot and have not been that comfortable at all.
"We found some good sectors in qualifying. Q1 and Q2 were very good.
"I just couldn't progress from there, the car was too difficult to drive in Q3. Too difficult to get the three tenths out of it compared to Carlos.
"But I am pleased with third. It looked like it could have been better but I think we finished where we should be."
Ferrari claimed a dominant one-two at the United States Grand Prix, with Leclerc taking victory, and Norris is wary of the pace of the Scuderia.
"The last few weekends they have been quicker than us. Today we were not on their level, but tomorrow is another day," Norris said.
"Ferrari have been the guys to be beat and Carlos is on top today. It will be challenging to beat them tomorrow.
"Carlos is going to be fast and has nothing to lose. We will try our best, but I don't think we have the pace compared to them at the minute."
Norris will not have support from team-mate Oscar Piastri after the Australian suffered a dramatic dip in form - being knocked out in the first part of qualifying having been fastest in final practice just hours earlier.
In a blow to McLaren's constructors' title bid, Piastri qualified only 17th and was joined in a Q1 elimination by Verstappen's team-mate Sergio Perez.
The Mexican, who admitted on Thursday he had endured a "terrible season", disappointed his raucous home support by finishing a lowly 18th as pressure intensifies on his Red Bull future.
McLaren lead Red Bull by 40 points, with Ferrari only a further eight points adrift.