Valencia triumphed 5-3 on aggregate to reach the UEFA Europa League semi-finals with Alcacer scoring a hat-trick for the home side.
The Spanish club recovered from a 3-0 first leg defeat in Switzerland, with Alcacer first two goals, plus a header from Eduardo Vargas, sending the match into extra time.
Basel pair Gaston Sauro and Marcelo Diaz both saw red before Alcacer put Valencia in front with seven minutes remaining in extra time. Juan Bernat then capped off the hosts' impressive performance in the 118th minute.
Pizzi thanked Valencia's supporters after the match for inspiring his players to produce their best.
"The atmosphere was crucial," Valencia's coach said.
"When one feels the atmosphere since stepping off the bus, it brings out the qualities of the players.
"That level of support is indispensible in these types of matches."
Alcacer dedicated the win to Valencia's fans, while he also paid tribute to Oscar Martinez, a local young footballer, who died of a heart attack.
"The fans have been with us right to the end. These fans are always there, through the good and the bad," the 20-year-old striker said.
"We would like to dedicate this win to them and most of all, Oscar Martinez, the youngster from Ontinyent, who is always in our minds."
Alcacer kick-started Valencia's comeback with a sharp finish from Joao Pereira's chipped pass in the 38th minute, while Vargas dragged the hosts within a goal of Basel three minutes before half-time.
In the 70th minute, Alcacer blasted the ball into the top corner from Vargas' knockdown to level the scores, while the striker put the La Liga outfit ahead with a tap-in from Pablo Piatti's cross, before Bernat scored on the counter-attack to complete the win.
"As long as I live, I'll never forget this hat-trick," Alcacer said.
"It's been my first in front of the Valencia fans.
"I'm very happy and I dedicate it to my team-mates, because without them, I couldn't have scored."
Pizzi was thrilled with Alcacer's performance.
"The centre-forward position is essential," the 45-year-old coach said.
"When a player scores three goals, there's very little else to say. The best thing he's done is to keep he and his teammates striving."
