Piastri won the 2021 Formula 2 Championship with Prema Racing
Alpine had lodged a claim with the board, a team of lawyers set up to adjudicate in contract disputes, after Piastri rejected their claim he was committed to them.
The CRB said: "The only contract to be recognised is the contact between McLaren and Piastri dated 4 July 2022. Piastri is entitled to drive for McLaren for the 2023 and 2024 seasons."
Alpine said they "acknowledged the decision", adding: "We consider the matter closed on our side and will announce our full 2023 driver line-up in due course."
Piastri, the reigning Formula 2 champion who has been Alpine's reserve driver this year, said: "I'm extremely excited to be making my F1 debut with such a prestigious team as McLaren and I'm very grateful for the opportunity that's been offered to me.
"The team has a long tradition of giving young talent a chance, and I'm looking forward to working hard alongside Lando [Norris] to push the team towards the front of the grid.
"I'm focused on preparing for my F1 debut in 2023 and starting my F1 career in papaya."
McLaren team principal Andreas Seidl said: "Oscar has an impressive racing career to date, and we are sure that together with Lando, he will be able to help us move another step forward towards our ambitions."
The decision ends a remarkable few weeks that began with Fernando Alonso's decision to move from Alpine to Aston Martin for 2023.
Alpine had been trying to juggle Alonso and Piastri, offering the two-time champion a one-year deal to stay with the team in 2023 while trying to farm Piastri out to Williams to gain experience.
But they have managed to lose both drivers - Alonso was frustrated by their unwillingness to offer a longer-term deal and their concern that he might lose performance, while Piastri did not appreciate not being offered the drive at the team he believed they ought to have provided.
It led to an extraordinary day after the Hungarian Grand Prix before F1's summer break in which Alpine said they were confident in their contractual hold over Piastri and later issued a press release saying he would drive for them in 2023, only for the 21-year-old to say on Twitter that he would not.
It became clear that Piastri's frustrations with Alpine had opened the door to McLaren, and it has now emerged that they signed a deal with him nearly two months ago.
Since then, McLaren have negotiated an early end to Daniel Ricciardo's contract to make way for Piastri to join Norris.
Ricciardo's future in F1 is uncertain, and Alpine are trying to secure Frenchman Pierre Gasly, currently tied to Red Bull-owned Alpha Tauri until the end of 2023, to partner Esteban Ocon.
Red Bull advisor Helmut Marko on Friday confirmed that talks are taking place with Alpine.
"We have a valid contract for 2023 but in the meantime there are talks," Marko told Sky Germany. "If our conditions are met, we wouldn't stand in Gasly's way. It would be a dream come true for him to drive in a French factory team.
"But not all the conditions have been met yet."
Asked who would be in the frame to replace Gasly amid speculation linking the team with IndyCar driver Colton Herta, Marko said: "We've already thought about that, but we don't want to announce that yet.
"We have some people in the junior squad who have a super licence. For the reserve driver, we would recruit someone from that."
-- Courtesy of BBC Sport