McLaren driver Piastri was placed just outside the podium places on Sunday as Briton Russell ended his 112-day wait for a race victory with Max Verstappen second, 1.6 seconds behind, with Italian Kimi Antonelli third to maintain his overall championship lead.
Piastri finished ahead of Lewis Hamilton in fifth, who was unable to build on his maiden win for Ferrari at the Barcelona-Catalunya Grand Prix last time out.
Charles Leclerc had started the race from second, with Ferrari team-mate Hamilton third, but they gradually dropped back as they lacked sufficient power to keep up with the leaders.
Asked if it was something of a surprise to move up to fourth, having started from seventh on the grid, Piastri admitted: "A bit, yeah. I think today was a really strong afternoon.
"I think it was a bit of a surprise that the Ferraris were not stronger, but on a personal level I thought it was a really good race.
"I think it was again a positive we're able to beat Ferrari – we've still got a lot to find to catch Mercedes and now Red Bull look very quick as well, so it's going to be a constantly changing order, I think."
Isack Hadjar, Verstappen's Red Bull team-mate, crossed in sixth with reigning world champion and Piastri's team-mate, Lando Norris, seventh.
Piastri now sits fourth in the drivers' championship on 80 points while Russell has moved back into second at the expense of Hamilton after his win in Austria.
Mercedes have extended their lead in the constructors' championship thanks to Russell's win and Antonelli's third-place.
"Yabba-dabba-doo!" a delighted Russell blasted into his microphone as he crossed the line, having been without a win since the opening fixture of the season at the Australian Grand Prix in Melbourne on March 8.
"Incredible to be back on the top step again," the Mercedes driver added, having controlled the race throughout and held off a challenge from Dutchman Verstappen.
"It has been a while so I am going to enjoy this one this evening, a lot of hard work with my team to get us back on track."
"I look forward to heading to Silverstone next week (British Grand Prix) and seeing the home fans. I am sure it is going to be great."
Championship leader Antonelli said: "I was a bit too excited in the first laps and I didn't drive well and made too many mistakes. The pace at the end was very strong so it was a shame I joined the party a little bit late."
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