Arsenal manager Mikel Arteta made just one change to the side that saw off Slavia Prague in the last round, with Martin Odegaard coming in for the injured Alexandre Lacazette.
The game, however, started in the worst possible way for the visitors as Trigueros broke the deadlock in only the fifth minute.
Things went from bad to worse for Arsenal when their defense failed to properly deal with a corner kick, allowing Albiol to convert Gerard Moreno's knockdown on the half-hour mark.
The Gunners were handed a lifeline when Juan Foyth brought down Nicolas Pepe inside the box. However, referee Arthur Dias Soares rescinded his initial penalty call for a handball in the build-up after consultations with the VAR.
Villarreal were undoubtedly the better side in the first half and deservedly went into the break with a two-goal lead.
Arsenal cranked up the pressure at the start of the second half, but their job was made more difficult when Dani Ceballos was sent off for two bookable offenses.
It saw the home side reclaim the initiative as they went close to scoring another goal through Moreno and Nigeria international Samuel Chukwueze.
Trigueros was then penalized for fouling Bukayo Saka and a penalty was consequently awarded to Arsenal in the 73rd minute.
Nicolas Pepe stood to half the deficit from 12 yards and the hosts soon received a red card of their own when Etienne Capoue was sent for a quick shower with 10 minutes to go.
Villarreal ultimately held on for the win ahead of the return leg at the Emirates next Thursday.
Here are five talking points from the enthralling fixture.
#5 Unai Emery gets one over his former side
Unai Emery was named Arsenal coach in the summer of 2018, but he was always going to have a tough task to follow in the footsteps of the legendary Arsene Wenger.
The former Valencia manager never truly settled in England and it came as no surprise when Arsenal sacked him 18 months after his appointment.
Since then, he has returned to his homeland to take charge of Villarreal and has taken the Yellow Submarines to new heights, while Arsenal are arguably worse off than when he left them.
When the Europa League draws were made, one of the sub-plots was a potential meeting between Emery and his former side, and the two clubs' respective quarter-final triumphs set the stage for a rekindled alliance.
Unai Emery would have relished the opportunity to get one over the Gunners and his excellent tactics on the night saw his side draw first blood.
With 90 minutes minimum still left to play in the tie, Emery's Villarreal have the advantage over Arsenal and the 49-year-old would be hoping to have the last laugh after the second leg.
#4 Nicolas Pepe's precious away goal gives Arsenal a lifeline
With just over 20 minutes to go in the game, Arsenal's hopes of progressing to the final hung by a thread.
For starters, they were already two goals and a man down, while the hosts were in the ascendancy and pushing for a third goal.
However, the Emirates outfit were handed a lifeline when referee Dias Soares pointed to the spot and Pepe made no mistake from 12 yards out.
It was a precious away goal for Arsenal and with their entire season dependent on the Europa League, Pepe's goal could be one of the most important scored by the Gunners this season.