Wes Brown's own goal 15 minutes from time ensured rock-bottom Sunderland remain seven points from safety.
Despite having two games in hand on many of their relegation rivals, their latest defeat leaves them four points adrift of the 19th-placed Cardiff City, who beat Southampton 1-0 on Saturday.
Poyet, who saw his side play well in patches and limit Everton - who made it seven league wins in a row - for much of the game, felt the result summed up Sunderland's dismal league campaign.
"We competed with one of the form teams of the division," he said.
"But if you need to show somebody somewhere in the world who doesn't watch Sunderland every week, you show them this and say this is the story of our season.
"That's it, you don't need to show them any more. Something always happens and sometimes, most of the time, that is against us.
"Whatever is it, it could be a decision, an own goal, a deflected shot. It is difficult to take, for sure.
"It hurts because no matter what we do, something happens... and it's not an excuse.
"We tried, we competed, we were very close to one of the form teams in the league but it was not enough.
"It's another day that we've let go past for whatever reason because of another own goal."
Sunderland's next two fixtures see them travel to Manchester City and Chelsea, and Poyet acknowledges time is running out for his team.
However, he took heart from Saturday's showing, adding: "When you play like that you always have a chance but we're running out of games and we need to make sure that it changes very quickly.
"As we know the next two games, they're going to be a little bit more difficult because of the quality of the opposition, so we'll see."
