Sebastian Larsson scored a fine 67th-minute free-kick to put Poyet's men on course for back-to-back victories following Monday's 3-1 triumph at Crystal Palace.
But Leighton Baines made sure it was honours even, scoring from the penalty spot nine minutes later after Connor Wickham's foul on Seamus Coleman.
"I think you need to take the point when you try and analyse football in a fair way," Poyet said.
"The opposition were very good on the ball and they controlled the tempo of the game so you need to adapt and be good when you get the ball.
"I think we created enough to win it but Everton also had their chances so overall I think it was one of those games – it was so tight that it could go either way so a draw was alright for everybody.
"The defence was the key. Everybody looks at the crosses you put into the box, the chances you create and the passes you make, but football is becoming more and more about how you defend and how much you care about defending.
"The team was practically perfect apart from a couple of actions and everybody needed to stand up, but we are working very hard at this point to show that we are back in business."
Sunderland's players reacted furiously when referee Lee Mason awarded the pivotal spot kick, however, their boss had no complaints despite expressing sympathy for Wickham.
He said: "It was difficult because Connor needed to try and stop the shot and sometimes you give the opportunity to the referee to give a penalty.
"I think a few people think that it wasn't but if I was the referee I would have given it, so overall I think it was quite fair."
