The Briton's third victory of the season cut Australian Piastri's Formula 1 lead from 22 to 15 points after 11 of 24 rounds, with the two McLaren drivers locked in an increasingly private title battle.
Charles Leclerc completed the podium for Ferrari -- who moved up to second overall -- but a hefty 17 seconds behind Norris, with teammate Lewis Hamilton fourth and another nine seconds down the road.
"Tricky, hot, tiring," said Norris after stepping out of the car at the finish of the 70-lap race in sweltering heat. "A lot of stress but a lot of fun. A nice battle, so well done to Oscar.
"Hopefully it was a nice one for everyone to watch but inside the car it was tough, especially when he was in DRS (drag reduction range)."
George Russell, winner in Spielberg last year and also in Canada two weeks ago, was fifth for Mercedes.
Red Bull's four-time world champion Max Verstappen suffered his first retirement of the season, in his team's home race, after a first-lap collision with Mercedes's Italian rookie Kimi Antonelli.
The Dutch driver stayed third overall, but is now 61 points behind Piastri and his hopes of a fifth successive title are fading fast. Antonelli was given a three-place grid drop for next weekend's British Grand Prix at Silverstone.
Piastri, who seized second place from Leclerc after an aborted first start, set the fastest lap of the race.
He took the lead momentarily on lap 11 in an intense battle but Norris grabbed it straight back in a duel that risked a repeat of the collision between the pair at the previous race in Canada.
Norris caused that one, and came away empty-handed, but this time it was Piastri who had the McLaren bosses shifting uneasily on the pitwall when he locked up with a puff of smoke on lap 20 while seeking to sneak through on the inside.
"Feedback from the pit wall, the manoeuvre in turn four with the lock up was too marginal. We can't do that again," Piastri's race engineer informed him over the radio.
'Please help'
The first set of pitstops took some of the heat off Norris, with Piastri staying out three laps longer, but the final stint brought them back together with the Australian looming in the Briton's mirrors.
With eight laps to go, the gap under two seconds and with a damaged front wing, Norris told his engineer Will Joseph that he needed some pace -- "please help", he said.
Piastri had already almost collided with Alpine's Argentine rookie Franco Colapinto who came dangerously close while being lapped and forced the McLaren on to the grass.
At the chequered flag just 2.695 seconds separated the McLarens, who celebrated their eighth win in 11 starts and fourth one-two of the season.
"I tried my absolute best. I probably could have done a better job when I just got ahead momentarily. But it was a good battle. A bit on the edge at times, and probably pushed the limits a bit far," said Piastri.
New Zealander Liam Lawson took a career-best finish in sixth for Racing Bulls with Fernando Alonso seventh for Aston Martin - the Spaniard finishing ahead of the Brazilian rookie he manages, Sauber's Gabriel Bortoleto.
The points were Bortoleto's first in F1 and to make it even sweeter Sauber took a double points finish, with Nico Hulkenberg finishing ninth.
Hulkenberg was the first Sauber driver since Valtteri Bottas in 2022 to score in three races in a row. Esteban Ocon took the final point for Haas.
Verstappen's teammate Yuki Tsunoda completed a dismal day for the former champions by collecting a 10-second penalty for a collision with Colapinto and being classified last.
Williams suffered a double retirement, with Alex Albon out after 15 laps and Carlos Sainz a non-starter.
McLaren are 207 points clear of Ferrari, who were without team boss Fred Vasseur who had to return home for personal reasons, in the constructors' championship