The strugglers conceded three goals during a lacklustre first-half display, Santiago Vergini gifting one to Arsenal striker Olivier Giroud, who scored twice, with a lazy backpass.
Sunderland improved after the break, but were unable to avoid a heavy defeat and remain in the Premier League relegation zone.
Poyet warned his side, who face Manchester City in the League Cup final next weekend, they must improve quickly if they are to retain their top-flight status.
"The combination of Arsenal playing well and us being poor on the ball made it difficult for us," he said.
"We needed to be organised and see the danger but they punished us.
"We were poor in many things and when you play a team like Arsenal you need to be spot on, but technically we were poor, (and) in terms of reading the game we were poor.
"We were second-best in reactions, sprints, power, in everything - and when that happens you lose, simple as that."
Sunderland had lost just two of their previous 16 games in all competitions prior to Saturday's match.
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