A sixth victory of the season for Norris was his first time back on top of the podium since winning the Hungarian GP at the start of August.
More important, Norris pounced on the recent slump Piastri has fallen into to tighten the teammates' bid to dethrone Max Verstappen as world champion.
The British driver holds a one-point advantage over Piastri with four races remaining.
Norris, who started from pole, opened Sunday's race at Autodromo Hermanos Rodriguez trailing Piastri by 14 points in the driver standings.
McLaren have already clinched their second consecutive constructors' championship and are now seeking to end Verstappen's four-year reign as F1's world champion with either of their drivers.
Norris is back on top of the standings for the first time since the fifth race of the season, in April.
"It's one weekend at a time," Norris said over a booming chorus of boos from the sold-out crowd of 150,000 spectators.
The booing caused Norris to pause his trackside interview, and he laughed nervously before continuing.
"I'm happy, I am focused on myself, I keep my head down, I ignore all of it,'' the Brit said.
"I keep to myself and it is working."
Piastri came from seventh and used a late pass of George Russell to finish fifth - a critical gain because it allowed Norris to take just a one-point lead over his Australian teammate in the championship race.
Verstappen arrived in Mexico City with three wins in the past four races to put himself back into title contention.
He finished third on Sunday, with a late virtual safety car preventing the Red Bull driver from passing Charles Leclerc for second.
Verstappen was 104 points behind Piastri six races ago, but has put himself in position to challenge for a fifth consecutive title. He is now just 36 points out of the lead.
The gap could have been even tighter had the safety car not limited Verstappen's chances to pass Leclerc over the final two laps.
"I mean, you win some, you lose some, right?" Verstappen said. "Sometimes the safety car works for you and sometimes it works against you."
Young British driver Oliver Bearman finished a career-best fourth to give Haas the second top-four finish in team history.
Norris's first career victory in Mexico City seemed like an easy Sunday drive as he crossed the finish line more than 30 seconds ahead of Leclerc.
"I could just keep my eyes focused, and just keep eyes forward and focus on what I was doing," Norris said.
"A pretty straightforward race for me, which is just what I was after."
Lewis Hamilton was in early contention for his first podium since joining Ferrari this year until a penalty ruined his race.
On the sixth lap, Hamilton went off-course in a three-wide race for third between Verstappen and Russell. He used an escape route on the course and rejoined the field back in third position, but was handed a 10-second penalty for leaving the track and gaining position on his return.
Ferrari have now gone an entire year without a victory since Carlos Sainz's win in Mexico City last October .
The next F1 race is the Brazilian Grand Prix on November 9. Verstappen has won the past two races at Interlagos.
