The Black Cats are now seven points adrift of safety at the foot of the table after going down narrowly to the Toffees at the Stadium of Light on Saturday.
Sunderland had more shots on goal than their Champions League chasing opponents, and more on target over the course of the 90 minutes too.
Fabio Borini had a shot cleared off the line by John Stones in the first half but the Cats were ultimately undone when an effort from a tight angle by Gerard Deulofeu deflected in off Wes Brown with a quarter of an hour remaining.
"It's harsh," Poyet told Sky Sports afterwards. "There have been four or five games since I came here that we couldn't win, that maybe we were very close to winning. That can change your season.
"It's the timings... sometimes you feel the defeats are coming in a terrible moment.
"But that is the story of the season. Something always happens that seems to go against you - it could be an own goal, a referee's decision or a deflected shot.
"When you're at the top, it goes for you. When you're at the bottom, it doesn't. And it hurts. It hurts a lot.
"That is the worst part in football. That shows you that when you're at your best, you can compete with any team in this league.
"But you need to be at your best yourself first and not make any mistakes and unfortunately, every single little mistake that we are making goes against us in a heavy way.
"But that's football. It's difficult to accept, but you have to try and the boys tried everything today."
Sunderland face tough away games at title contenders Manchester City and Chelsea before their next home game against relegation rivals Cardiff at the end of the month.