Passing pole-sitter Leclerc on the 20th lap in Baku, the 23-year-old Australian defended lap after lap as the Ferrari sought a way past.
A big collision between Ferrari's Carlos Sainz and Red Bull's Sergio Perez on the penultimate lap assured the win with a virtual safety car finish.
"For me, I think that ranks as probably the best win of my career, I would say. Trying to soak up that pressure for so long in that race, that was incredibly tough," said Piastri of his second career F1 victory.
"That was probably the most stressful afternoon in my life," he added.
Piastri's victory came after McLaren boss Andrea Stella had said the team wanted to support team mate Lando Norris in his title fight with Red Bull's triple world champion Max Verstappen.
On Sunday, in a role reversal, Norris was asked to help his team mate by holding up Red Bull's charging Sergio Perez and creating a window for the Australian to pit for fresh tyres and stay ahead.
"Can you do anything to hold Perez up without hurting yourself?" Norris, who had yet to pit and was immediately ahead of Perez, was asked by his engineer.
The teamwork succeeded with Piastri staying second after his stop, which allowed the Australian to lunge past Leclerc on his way to becoming the top scorer on the grid over the last seven rounds.
"I think I have had a good run through the European part of the year and now continuing, but... yeah, it's not just down to me," said Piastri.
"We had a car that could put us in the fight. We had a pit stop that could put us in the fight. We had some teamwork that put us in the fight. And it all managed to pay off."
Norris, who finished fourth after starting 15th and also managed to cut Max Verstappen's overall lead to 59 points, said he had "played the team game well" and that was what made him the "happiest".
"I did my small part for the team, which I'm very happy for because it got us to P1 in the constructors'," he told reporters.
McLaren have consistently said they do not view any of their drivers as primary or secondary.
Red Bull boss Christian Horner suggested having two drivers racing on equal terms could backfire, particularly when "the second driver starts out-performing the first driver".
"Obviously, at the beginning of the season, it's all open, but certainly when you get to around the halfway point, you've got to pick a horse, especially if you're in a championship battle," he said.
