The mood of visiting manager Sam Allarydce won't have been lifted by the absence of defender Michael Keane, who is out with a gashed foot, and influential midfielder Idrissa Gueye (hamstring), although Cenk Tosun, the £27m signing from Besiktas, will make his debut for the club as he has been named in the starting XI.
Spurs, chasing a top-four finish for the third successive campaign, went into the game having tasted defeat just once in their previous ten league outings. A player the Toffees chief labelled - the best in Europe - at that price.
Son's flicked header off an Eriksen free kick in the 14th minutes was the first sight of goal for either side but it flashed over Jordan Pickford's crossbar before Rooney was put in by Tosun's flick but he dragged a shot across goal and wide.
Indeed, the visitor's menace was very real.
Injuries: Antonio Brown leaves practice with illness
Brown set an National Football League record with 100-plus receptions in five consecutive seasons, including 101 this season. Brown suffered the calf injury early in the December 17 loss to New England and missed the two subsequent games.
Spurs should be good enough to secure a comfortable win here, and given Everton's struggles in front of goal, I can't see them getting past Hugo Lloris. But two minutes after the restart, he equalled Sheringham's haul and doubled Spurs' advantage.
Tottenham are without Harry Winks, Danny Rose and Toby Alderweireld but are expected to be unchanged from their last league outing.
The busy Pickford denied Son and Dele Alli in quick succession as Spurs threatened to run riot.
He added: "It will be a tough game to play because of the quality of their side, but you hope Wembley will bring the best out of our players because they have the chance to play in such an iconic stadium".
Their lack of activity has come in stark contrast to most of the Premier League's top six with Manchester City and Manchester United scrapping over Arsenal's Alexis Sanchez while Liverpool have already bought Virgil Van Dijk and sold Philippe Coutinho.