The French world champion Antoine Griezmann provided a double to equalize (28th) and a little later to lead the guests (35th).
Samu Chukwueze (26th) had put Villarreal in the lead.
Manu Trigueros saw the red card for a foul on Barça star Lionel Messi (65th).
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Klopp says frustration: “need time to be angry”