The in-form frontman struck midway through the first half to put United ahead but Richarlison levelled shortly before the break.
Rafa Benítez made three changes to his starting XI from Saturday's defeat to West Ham, with Jamaal Lascelles, Jacob Murphy and former Everton loanee Christian Atsu coming in for Ayoze Pérez, the injured Kenedy and the suspended Matt Ritchie.
That meant a return to the three-man central defence which served United so well at Burnley nine days previously and in the opening exchanges, the home side were restricted to a Cenk Tosun shot, which was deflected wide, and an André Gomes strike which flew well over Martin Dúbravka's crossbar.
Then in the 19th minute, Federico Fernández sent Murphy scampering away down the left. The wideman burst past Seamus Coleman before driving a magnificent low ball across the six yard box, which Rondón smashed home at the far post.
For a while afterwards, United looked relatively comfortable but Everton started to pile on the pressure in the 36th minute, when Fernández make a superb last-ditch block to thwart Gylfi Sigurdsson, who looked a certain scorer.
Moments later, Dúbravka flew to his left to somehow keep out a shot from Richarlison but, from the resulting corner - taken by Sigurdsson - the Toffees' Brazilian forward forced home from close range.
On the stroke of half time, Dúbravka made another great save, this time to deny Tosun after the Turkish striker met Lucas Digne's cross.
In the 59th minute, Atsu turned just inside the box and hit a powerful strike towards goal, but it was too close to Jordan Pickford then in the 75th minute, Fabian Schär saw a looping header saved by the Everton and England goalkeeper.
Everton, meanwhile, struggled to break down a well-organised Magpies defence and indeed it was nearly the visitors who won it, with Atsu squandering two good late opportunities against his old club.