The Toffees were the victims of their own downfall at times as they were dismantled by second half goals from Harry Kane and Christian Eriksen, with man of the match Heung-min Son opening the scoring in the first period.
Kane also became Spurs' all-time leading scorer in the Premier League, with his brace taking up to 98 league goals.
The 24-year-old has now moved past Teddy Sheringham, who netted 97 goals for the North London club in 236 matches.
However, after his countryman Cenk Tosun signed for Everton, the one-time Newcastle player once against couldn't resist getting a dig in.
He said, "I think the defining moment in the game was we thought the second goal was offside, probably the same with Wayne's".
But with 26 minutes gone, Son timed everything right when Serge Aurier's scuffed shot fizzed across the box and he steered the ball home from close range.
But today belonged to Harry Kane who brought his goalscoring total in the Premier League this season to 20. Sure. For me his quality, mentality and professionalism is top. "He is always thinking to try to improve, improve, improve, score goals and improve for the team". 'Hopefully his ability and goalscoring ability will help us along the way, but we've got to get back to the basics defensively like I did when I first started'.
Rooney's late booking for an ugly chop on Jan Vertonghen encapsulated Everton's frustration and Tottenham polished off a handsome win with a stunning fourth goal, Dele Alli's back-heel teeing up Eriksen to hammer emphatically past Pickford.
Sam Allardyce: "Maybe I have a bit of responsibility by playing too much attacking-minded players and not as many defending-minded players". I saw some good signs of the players last week against Liverpool and I saw some of the worst signs in the second half today which is massively worrying.
Alan Pardew oversaw his first Premier League victory as West Bromich Albion manager thanks to a 2-0 win over Brighton, while Rafael Benitez's Newcastle came from behind to draw 1-1 with bottom-of-the-table Swansea City as the battle to avoid relegation intensified.