The Portuguese champions trailed at the break but an inspired second-half display put Porto on the brink of a place in next month's final in Dublin.
"It's the first time I've scored a 'poker', four goals. A magical night that I will never forget," Falcao said.
"It was a great win. We didn't fall apart after going behind and were able to react rapidly."
Villarreal were dangerous in the first half and were rewarded in the closing seconds when Brazil forward Nilmar crossed for midfielder Cani to head the La Liga side in front.
The home side equalised on 49 minutes when Falcao converted a penalty after he rounded Diego Lopez and forced the keeper to bring him down.
Buoyed by the goal, Porto drove forward and went ahead on the hour-mark when Falcao's compatriot Fredy Guarin dribbled past Bruno Soriano on the right wing, cut in to fire a fierce shot against the post and headed in from the rebound.
Falcao added Porto's third in the 67th minute with a tap-in after Hulk led a counter-attack and then the fourth with a diving header from a curling free-kick by Guarin that left the Spanish defence standing.
The striker took his Europa League goal tally to 15 with a looping header from a James Rodriguez corner in the last minute.
"He's outstanding, a fantastic striker," said Porto coach Andre Villas Boas, adding that major European clubs who want to lure the Colombian away from Portugal will have to pay a big price tag.
