The Mexico international struck twice in the second half at Anoeta to give the Liga side a 4-0 aggregate victory and send them into Europe's premier club competition for the first time since 2003-04.
Jagoba Arrasate made two changes to the Sociedad side that won in France last week, as Daniel Estrada and Ion Ansotegi replacing Carlos Martinez and Liassine Cadamuro.
Meanwhile, his opposite number Remi Garde opted to select Samuel Umtiti ahead of Steed Malbranque, while Bakary Kone came in for the suspended Milan Bisevac, who was sent off in the game at the Stade de Gerland.
Lyon began the game brightly and dominated possession in search of an early goal to get themselves back in the tie.
However, they failed to test Claudio Bravo in the Sociedad goal and it was the home side who carved out the first clear-cut chance of the game.
Antoine Griezmann, who lit up the first leg with a sublime scissor-kick, raced through in the eighth minute, but fluffed his lines as his weak attempted chip was easily gathered by Anthony Lopes.
That opening gave Arrasate’s men some impetus though and Griezmann began pulling the strings.
Vela could only fire wide after the Frenchman cut a corner back to him just before the half-hour mark, before Griezmann himself had another effort saved by Lopes.
The former Arsenal man was then involved in the major talking point of the first half as he went down in the area under a challenge from Kone, only for referee Damir Skomina to book him for simulation.
And the 24-year-old could have made amends by opening the scoring a minute later but Lopes palmed his lofted shot wide.
Sociedad picked up where they left off after the break and Haris Seferovic almost grabbed his second goal of the tie on 49 minutes when he collected Vela’s pass, cut inside from the right and rifled a left-footed effort off the inside of the near post with Lopes beaten.
And Seferovic missed a gilt-edged opportunity to wrap up the home side’s victory shortly after as a swift counter-attack ended with Griezmann laying the ball across to him, but the Switzerland international could only side foot wide.
However, Sociedad did not have to wait long for the breathing space they were after as Vela put them three goals to the good on aggregate after 66 minutes.
Griezmann was again the creator as the former Arsenal forward met his inswinging corner with an unmarked header from inside the six-yard box.
And Vela saved his best for last, as he grabbed his second of the night with a brilliant solo effort in injury time that involved him beating two defenders before coolly lifting the ball over Lopes.
