Moreno opened the scoring for the Yellow Submarines against the run of play as Manchester United failed to find their rhythm early in the game and the first half ended in favour of Villarreal.
Edison Cavani restored parity for Manchester United after the restart.
Rashford missed a glorious chance to put Manchester United ahead.
The match could not produce any goal as both sides ended the extra time even.
The penalty shoot out was a tense as affair with both teams scoring the first ten penalties before David De Gea missed the decisive one to hand Villarreal the trophy.
“It’s a disappointed dressing room. That’s football for you. Sometimes it’s decided on one kick — and that’s the difference between winning and losing,” said United boss Ole Gunnar Solskjaer.