Emery, who was dismissed by the Gunners in November 2019 after a poor run of form, watched on as Villarreal seized control thanks to first-half goals from Manu Trigueros and Raul Albiol.
Dani Ceballos' sending off only appeared to strengthen the host's hand, but Nicolas Pepe's spot-kick in the 73rd minute breathed fresh life into a contest that looked in danger of getting away from Mikel Arteta's team.
Former Tottenham midfielder Etienne Capoue also saw red in a tie that is delicately poised ahead of next week's return meeting.
Villarreal scored with the first effort of the game, Trigueros drilling home after Samuel Chukwueze's jinking run ended with the ball being tackled into his team-mate's path.
Shoddy Arsenal defending saw the lead doubled just prior to the half-hour mark when Gerard Moreno flicked on Dani Parejo's corner for Raul Albiol to volley home from close range.
Juan Foyth was excelling going forward but encountered problems against Pepe defensively and only a VAR check for handball in the build-up saved the defender from a penalty call going against him for a trip in the area.
Arsenal started the second half on the front foot but their outing worsened when Ceballos, already on a yellow card, was dismissed for a late challenge on Parejo.
Bernd Leno was called into action to deny the excellent Moreno and that moment proved even more pivotal when Trigueros clumsily tripped Bukayo Saka, Pepe's penalty sent down the middle and just past Geronimo Rulli.
There was even more room for optimism when Capoue hacked down Saka and was dismissed while leaving the field on a stretcher, but Villarreal clung on for a precious win despite returning captain Pierre-Emerick Aubameyang testing Rulli deep into added time.