Newcastle, who have failed to score in four of their 10 league games this season, have now lost 10 consecutive top-flight matches on a Monday.
Rafa Benitez says Magpies had control until goal at Turf Moor.
Newcastle seemed happy to settle for a point and only showed any real urgency once they fell behind after Perez had carelessly lost possession.
"We had the game under control," said Magpies boss Rafael Benitez. "I'm really disappointed."
The visitors twice went close to equalising in the closing moments - Pope keeping out substitute Isaac Hayden, before Javier Manquillo's flicked header from Jonjo Shelvey's cross landed on the roof of the net.
On another Monday night to forget for the Magpies, Christian Atsu was guilty of wasting the best chance of the first half when he poked over from close range.
Asked what his side is missing, Benitez added: "Be more precise on the ball and be more clinical. We made two or three crosses that were not the best when we were in a very good position.
"We have to carry on and improve."