The struggling Wearsiders conceded three goals during a lackluster first-half display, Santiago Vergini gifting one to Arsenal striker Olivier Giroud with a lazy back pass.
Sunderland improved after the break, but was unable to avoid a heavy defeat and remains in the Premier League relegation zone.
Poyet warned his side, whose next fixture is against Manchester City in the League Cup final, that it must improve quickly if it is to retain its top-flight status.
"The combination of Arsenal playing well and us being poor on the ball made it difficult for us," Poyet told reporters. "We needed to be organized 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 its previous 16 games in all competitions prior to Saturday's match, but the defeat leaves the club in 18th place, a point away from safety with 12 fixtures to play.