It completed a good night for Spain as 10-man Sevilla tormented a fragile Porto side while competition favourites Juventus and Benfica also safely negotiated their quarter-finals.
Basle, who were reduced to nine men in extra-time, went into the second leg 3-0 up and full of confidence at a highly-charged Mestalla stadium, but they came up against a Valencia side with the bit between their teeth.
No team had overhauled a three-goal deficit in a knockout tie since the Europa League started in 2009 but the hosts signalled their intent when Alcacer beat keeper Yann Sommer after collecting Joao Pereira's pinpoint pass in the 38th minute.
The impossible became faintly probable four minutes later when Eduardo Vargas headed home from Fede Cartabia's corner and the deficit was then wiped out with 20 minutes remaining as Alcacer rifled the ball into the top corner to send the home fans into raptures and set up a thrilling finale.
Alcacer missed a superb chance to win the tie in normal time but was denied by the excellent Sommer when one-one-one and the extraordinary match went into extra time.
Basle's task was made even harder when Marcelo Diaz was then sent off for kicking out and Gaston Sauro was dismissed after picking up two yellow cards in two minutes.
The almost inevitable winner arrived with six minutes of extra time remaining when Alcacer steered the ball home from close range after Pablo Piatti's cross and Juan Bernat made the tie safe by lifting the ball over Sommer when through on goal.
"The fans were with us to the end and it's an honour to have this support in the good times and the bad," Valencia's hat-trick hero Alcacer said.
"I missed a chance late in the game and I was really down on myself but then thank God I had another in extra time and I was able to put it away."
- Sports & Recreation
- Yann Sommer
- Europa League