Manchester United drew the match but go through to the final 2-1.
The English side took a goal advantage into the second leg thanks to Marcus Rashford's free-kick and the 19-year-old was involved again as he set up Marouane Fellaini, who headed past a despairing Sergio Alvarez.
Celta equalised on the night with five minutes to go through Facundo Roncaglia, before both he and United Eric Bailly were given their marching orders late on in a dramatic finale.
In France, Lyon failed to find the crucial goal that would have seen them play Mourinho's men in the final. Some slack defending allowed Facundo Roncaglia to head the visitors level with just four minutes to go.
United started the second half much better than the first and nearly got a second goal when Henrikh Mkhitaryan's fierce shot was brilliantly saved by Sergio Alvarez.
"The boys didn't play a phenomenal match but they fought, they gave everything they had to give and I'm really pleased for them because 14 matches and we're in the final".
Ajax midfielder Donny van de Beek curled a shot against the angle of crossbar and post, which would have finally killed off the hosts, and they were left to face a frantic last 10 minutes when Ghezzal's header deflected in off Viergever, who moments later was sent off for a second bookable offence.
"It's obviously disappointing especially as we managed to win".
Man United stumbles into Europa League final
United have won the European Cup or Champions League three times, and the now-defunct European Cup Winners' Cup once. The England worldwide will miss the first three games of next season's Champions League campaign as a result.
Mourinho could end his first season at United with two trophies - he has already won the League Cup - and a place in the Champions League, which is guaranteed for the victor of the Europa League. They deserved their equaliser and kept passing and probing while their Ugandan-born forward, Pone Sisto, forced Sergio Romero to tip his shot into the Stretford End.
Romero; Valencia, Bailly, Blind, Darmian; Fellaini, Herrera, Pogba; Lingard (Rooney, min. 89), Mkhitaryan (Carrick, min. 77), Rashford (Smalling, min. 89).
When it came to European football, the great helmsman dealt mainly in Champions Leagues.
Celta Vigo (4-3-3): Alvarez; Mallo, Cabral, Roncaglia, Jonny; Wass (Jozabed ht), Radoja (Bongonda 68), Hernandez; Aspas, Guidetti, Sisto (Beauvue 80).
Jeered by Manchester United fans following his introduction in the 1-0 win over Tottenham at Old Trafford, just a week after conceding a late penalty that denied his side a win at Everton, Fellaini's future was starting to look bleak - and not for the first time.
It has been 21 years - the 1996 Champions League final - since Ajax was within one win of capturing a European trophy.
Substitute Rachid Ghezzal's heavily deflected header nine minutes from time made it 3-1 on the night and Dutch global Nick Viergever's sending-off gave Lyon extra impetus.
And certainly the opening for minutes it was the visitor's, playing their first European semi- final, who made all the running helped by some sloppy Utd defending.
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