Villarreal clinched the final place in the last 16 of the Champions League on Thursday, while Leicester failed to advance in the Europa League.
Villarreal beat Atalanta 3-2 away in a game postponed from Wednesday because of heavy snow in northern Italy.
The Spanish side only needed a draw to advance as runner-up from Group F but jumped out to a 3-0 lead by the 51st minute, with Arnaut Danjuma scoring twice. Atalanta answered with goals from Duvan Zapata and substitute Ruslan Malinovskyi but it was too late to turn things around.
Atalanta finished third in the group and will enter a playoff to get into the last 16 of the second-tier Europa League, the competition Villarreal won last season.