Fans vented their fury at the Uruguayan as Villa scored all four goals in a humiliating first half for the Wearside club at the Stadium of Light.
Sunderland are just a point above the relegation zone after winning only four top-flight games this season and Poyet's 17th-month tenure came to an end on Monday, with the club saying that an announcement on his replacement will be made "in due course".
In a statement released on Sunderland's official website, chairman Ellis Short commented: "I would like to thank Gus for his endeavours during his time at the club, in particular last season's 'great escape' and cup final appearance, which will live long in the memory of every Sunderland fan.
"Sadly, we have not made the progress that any of us had hoped for this season and we find ourselves battling, once again, at the wrong end of the table. We have therefore made the difficult decision that a change is needed."
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