Gerard Moreno opened the scoring in the first half of a poor final in Poland on Wednesday to move level with Giuseppe Rossi as the Yellow Submarine's leading all-time goalscorer with 82 - 30 of which have come this season.
Edinson Cavani equalised early in the second half with his sixth goal in five Europa League games for United and the two sides remained level at 1-1 after extra time.
There had been an astonishing 21 successful spot-kicks until United goalkeeper De Gea's tame effort was kept out by his opposite number Geronimo Rulli, with Villarreal winning the shootout 11-10.
Villarreal go into the Champions League next season after winning their first major European trophy, with head coach Unai Emery having lifted this trophy an incredible four times.
Ole Gunnar Solskjaer, on the other hand, remains without a trophy during his tenure after a flat performance from United.