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.
Basel's midfielder Taulant Xhaka (L) vies with Valencia's midfielder Fede Cartabia. Photo by JOSE JORDAN/AFP/Getty Images
Juventus kept their dreams of a domestic and European double alive with a 2-1 victory over Lyon.
After a 1-0 victory in the quarter-final first leg in France last week, the Turin giants secured a 3-1 aggregate victory to progress to the last four of a European competition for the first time since the 2002-03 campaign.
Inspirational midfielder Andrea Pirlo struck the opening goal after four minutes with an exquisite free-kick from just outside the penalty area, but the visitors levelled when Jimmy Briand placed a header beyond Gianluigi Buffon 14 minutes later.
The Italian champions were not to be denied, though, and sealed victory on the night thanks to a long-range Claudio Marchisio strike that took a wicked deflection off Samuel Umtiti and gave goalkeeper Anthony Lopes no chance.
Juventus' midfielder Andrea Pirlo fights for the ball with Lyon's French midfielder Steed Malbranque. Photo by OLIVIER MORIN/AFP/Getty Images
A stunning first-half performance helped Sevilla thump Porto 4-1 on Thursday to secure a place in the semi-finals.
Trailing 1-0 from the first leg, the home side wasted no time in overturning the deficit, stunning Porto with three goals in the opening 29 minutes to kill off the tie.
Finally, Benfica remain on track to avenge last season's UEFA Europa League heartbreak as they moved into the semi-finals with a 2-0 win over AZ.
Benfica's Spanish forward Rodrigo Machado (R) vies with Alkmaar's defender Nick Viergever. Photo by FRANCISCO LEONG/AFP/Getty Images