Pablo Alcacer scored a hat trick in Valencia, with the third coming in extra time after Basel's Marcelo Diaz and Gaston Sauro were sent off.
Eduardo Vargas and Juan Bernat also scored to help Valencia become the first side to come back from a 3-0 first leg defeat and progress in the Europa League.
"I'd like to thank my teammates, without whom I wouldn't have scored this hat trick tonight," Alacer said. "The fans were with us from the very first minute tonight; they were fantastic and we dedicate this win to them."
In Turin, Andrea Pirlo and Claudio Marchisio scored either side of Jimmy Briand's leveler for Lyon, handing Juventus a 3-1 aggregate win. Marchisio's strike was later awarded as an own goal by UEFA, after it took a decisive deflection off Samuel Umtiti.
"We hope that the whole of Italy is happy with our result," Juventus coach Antonio Conte said. "In Italy you're either a Juventus fan or against Juventus. And those who are against us don't care a lot about the UEFA ranking and only hope that we are knocked out.
"We are proud and happy. Three years ago we didn't even qualify for the Europa League, in the last two years we have done a lot and won a lot and this year we are protagonists too. The fans shouldn't forget that."
Benfica, beaten by Chelsea in last season's final, could be back in the showpiece match next month after beating AZ Alkmaar 2-0 to seal a 3-0 aggregate victory, while 10-man Sevilla defeated Porto 4-1 to qualify 4-2 on aggregate.
Basel had seemed certain to progress after a comfortable win in Switzerland, but two goals shortly before halftime gave Valencia hope.
First Alcacer chested down Joao Pereira's ball over the defence and fired it into the bottom left corner, before Vargas headed home Dani Parejo's corner.
The two goalscorers combined in the 70th minute with a delightful cut back from Vargas to the onrushing Alcacer, who lashed the ball into the top left corner to level the tie.
Basel goalkeeper Yann Sommer kept his side in the tie with a number of stunning saves, while counterpart Vicente Guaita saw little of the ball.
Valencia almost won it on the stroke of full time but Sommer did brilliantly to deny Alcacer and Vargas slid the rebound off the base of the right post.
Basel lost hope of scoring a crucial goal when Diaz was shown a straight red card for a needless kick at Ricardo Costa and Sauro followed him five minutes later after two yellow cards in a minute.
Alcacer completed his hat trick when he tapped home Pablo Piatti's cross seven minutes from time and Bernat sealed the result five minutes later.
Juventus is favourite to win the competition, with the final in its own stadium on May 14, and it looked dangerous from the start as Carlos Tevez threatened to break his four-year goal drought in Europe with a shot which flew narrowly over the crossbar.
The Italian side did take an early lead when free-kick specialist Pirlo curled an effort into the left of goal from just outside the area.
Lyon was denied what looked like a clear penalty when Juventus defender Leonardo Bonucci blocked Alexandre Lacazette's cross with his hand but the referee waved play on.
Lyon tied it when a corner came out to Arnold Mvuemba, who sent a looping cross into the area for Briand to outjump Marchisio and head in off the base of the far post.
Tevez thought he had finally broken his drought when he headed home Pirlo's cross on the hour mark but it was ruled out for offside.
Lyon had looked the more dangerous but Juventus took the lead in the 68th minute as Marchisio's shot took a huge deflection off Umtiti to completely wrongfoot Anthony Lopes.
Sevilla was trailing 1-0 from the first leg but levelled the tie in the fifth minute after Danilo brought down Carlos Bacca and the Colombia striker drilled the resulting penalty into the bottom right corner.
Vitolo and another strike from Bacca gave Sevilla a comfortable advantage before defender Coke was sent off for two bookings less than 10 minutes into the second half.
However, substitute Kevin Gameiro sealed Sevilla's progress when he fired home a rebound in the 79th minute before Ricardo Quaresma notched a stunning consolation for Porto in stoppage time.
In Portugal, Rodrigo scored in either half to keep Benefica on course for a second consecutive final.