Vela, 19, had spent the whole of this season on loan at Spanish side Osasuna as the Mexican striker awaited clearance.
Now he has been given the green light to play for the Gunners in 2008/09 and will link up with the Premier League club for pre-season training in July.
Vela, who joined Arsenal from Celta Vigo in January 2007, has already made three appearances for the full Mexican national team, scoring once.
Arsene Wenger has high hopes for Vela and recently hinted he could fill in for injured Croatia forward Eduardo.
"Vela will be involved with the first-team next season. He has played the whole season at Osasuna on the left wing and he had to work hard there," Wenger said. "But for me it is not his position, he is a central striker, an Eduardo type. He is a great young player."