A 25th-minute penalty from Ki Sung-Yueng gave the Wearside outfit their first away win of the season in the top flight.
Ki's penalty was the turning point as Tim Howard was sent off for bringing down the Swansea City loanee.
Everton threw everything at Sunderland, but they were unable to maintain their unbeaten home record.
Sunderland remain bottom of the table, but Poyet was delighted with the three points and also claimed Everton are the best team his side have faced this season.
"For us, three points is something incredible," he said.
"I say it before but Everton are the best team we have played against.
"Even with 10, I have no doubt."
Poyet also praised goalkeeper Vito Mannone, who turned in a stunning second-half display to keep Everton at bay.
"Sometimes the strikers get all the credit but sometimes you need the goalkeeper to win you the game and Vito won us the game for us," he said.
The Uruguayan was at a loss to explain why his side, who have now beaten Everton, Manchester City and Newcastle United this season, are only able to beat those in the upper reaches of the division.
He added: "I have no explanation for that. How? Why? We can not blame anyone because we had the chances to beat those around us but always we do something wrong.
"We need to change that if we want to stay up.
"When we play against the top teams it goes the other way, we defend, the ball doesn't go in, the goalkeeper is man of the match."
