No side had ever come from three goals down to win a knockout tie since the advent of the Europa League, but Valencia did just that, triumphing after extra-time courtesy of a 5-0 second-leg win on an extraordinary night at the Mestalla.
Basel boss Murat Yakin had captained the Swiss side in a 6-2 defeat at the same venue 12 years ago and could only watch on in dismay as the visitors, who finished with nine men, imploded.
Alcacer produced a cool finish seven minutes before half-time to give Valencia hope and Edu Vargas made it 2-0 on the night with a powerful header four minutes later.
Both strikers struck the woodwork after the break as Valencia desperately sought to level the tie, eventually doing so in the 70th minute through Alcacer.
Alcacer should have scored again towards the end of 90 minutes, but couldn't convert a one-on-one chance, before Vargas struck the post with the rebound.
An under-pressure Basel then saw Marcelo Diaz and Gaston Sauro see red in the first half of extra-time, and Valencia took full advantage, with Alcacer completing his hat-trick after 113 minutes and Juan Bernat rounding off a remarkable win with a late fifth.
Valencia, who were without Diego Alves, Philippe Senderos and Victor Ruiz, initially struggled to find a way through a resolute Basel defence that had only conceded once in their last five Europa League games.
A frustrated home crowd were brought to life seven minutes before the break, however, when a nicely-weighted pass from Joao Pereira found Alcacer, who beat the offside trap before controlling the ball superbly with his chest and stroking home a cool finish from close range.
That strike clearly gave the hosts a fresh injection of confidence, and the lead should have been extended four minutes later when Vargas' point-blank header was kept out by Yann Sommer.
Vargas made amends for that miss from the resulting corner, rising highest to meet Daniel Parejo's delivery with a towering header into the top-right corner.
The Chilean should have been celebrating again nine minutes after the break when he burst through the heart of the visitors' defence to force an inspired save from Sommer, before clattering the rebound off the post from a tight angle.
Sommer was called into action again on 66 minutes, tipping Alcacer's curling effort from 20 yards onto the crossbar.
But he was beaten four minutes later when a hopeful long ball was brought down by Vargas. He set up Alcacer on the edge of the area, who lashed home an unstoppable shot via the underside of the crossbar.
Alcacer should have completed the comeback in the 88th minute but was denied by Sommer, before Vargas slid in to strike the post again.
Basel were yet again thanking Sommer within the first three minutes of extra-time, the goalkeeper this time denying Parejo after some good work down the left from Bernat.
The visitors then lost their composure and Diaz saw red for an off-the-ball incident before Sauro picked up two bookings in a minute.
Alcacer duly converted a low cross from substitute Pablo Piatti at the near post to give Valencia the lead and a famous win was completed in style when Bernat found himself on the edge of the box, before producing a cool finish to secure his side's place in the semi-finals.
