It was a jointed display from Arsenal, who started with new signing Viktor Gyokeres up front for the first time, but the striker was unable to make much of an impact.
Nicolas Pepe scored Villarreal's opener after a quarter of an hour as he came back to haunt his former employers, hitting the post from range before sprinting into the box and capitalising on some remarkably poor defending from Jakub Kiwior, who attempted to stop the winger reaching the ball and roundly failed.
Just after the half-hour mark, the Spanish side doubled their lead through Karl Etta Eyong, as he hooked a volley into the net after David Raya failed to claim an effort from an angle. However, within two minutes Arsenal had a goal back through Christian Norgaard, thanks to a smart header from a corner.
But despite territorial dominance, the Gunners conceded again on 68 minutes, with ex-Tottenham loanee Arnaut Danjuma bursting clear of the offside trap and finishing low into the bottom corner. Seven minutes later Arsenal won a penalty, as substitute Dowman slalomed his way into the penalty area and was brought down, and Martin Odegaard converted from 12 yards.
Mikel Arteta's side poured forward in an attempt to equalise, but were unable to create a clear-cut chance. After the game, the two sides conducted a penalty shootout. Mikel Merino, Gabriel and Noni Madueke missed for Arsenal.