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 However, referee Arthur Dias Soares rescinded his initial penalty call for a handball in the build-up after consultations with the VAR.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.
#3 Dani Ceballos allows frustration to get the better of him
Dani Ceballos started the game at the heart of Arsenal's midfield but a superb display by Dani Parejo and Etienne Capoue meant the hosts dominated proceedings from the middle.
This saw Ceballos and his midfield partner Thomas Partey frustrated early on, but it was the Real Madrid loanee who lost his cool.
A series of rash tackles were made before his booking in the 44th minute, but rather than towing a safer path in the second half, the 24-year-old continued to live life on the edges and he was deservedly sent off for a stamp on Parejo.
His midfield partner Partey was also shown a yellow card for protesting the decision and Ceballos' dismissal meant that Arsenal played the final 33 minutes of the game with just 10 men.
#2 Balanced scoreline means there is all to play for in the second leg
As already indicated, Arsenal's away goal gives them a massive lifeline heading into the second leg, but Villarreal will still fancy their chances against a Gunners side that have been plagued by inconsistencies.
Villarreal themselves have not exactly set the world alight with their displays this season and the tie is still very much in the balance.
The defeat means Arsenal must secure a win on home soil next week, while Villarreal just have to avoid losing at the Emirates to progress to the final.
Both sides have what it takes to get the job done in the second leg and the finely balanced scoreline means that another exciting game could be on the cards in the English capital.
#1 Impressive Bukayo Saka steps up when it matters most for Arsenal
Bukayo Saka has arguably been a revelation for Arsenal and having had his breakout campaign last term, he has continued his impressive trajectory in the current season.
With Pierre-Emerick Aubameyang and Alexandre Lacazette missing through injury, the onus was on the 19-year-old to lead the line and he did so with minimal fuss.
Even as Arsenal struggled to clear their lines, the England international continued to charge forward and put Villarreal's defense on the backfoot.
His efforts reaped dividends when he won the penalty from which the visitors scored. He also drew the foul from Capoue that got him sent off and saw Arsenal play the final 15 minutes at numerical par with their hosts.