Christian Eriksen rounded off the scoring for the host in the 81st minute. The Toffees have bagged a single goal in their last five matches; combine that with a single win in 20 away outings and you can see why Everton supporters are aging at a disproportionate rate.
Harry Kane will be hoping to follow up his two goals against AFC Wimbledon with more against a struggling Everton side.
Son, who has had a hand in more goals at Wembley than any other Tottenham player this season with seven goals and four assists, was comfortably the best player on the pitch but it was Kane who assumed centre stage as his 98th top-flight goal arrived in the 59th minute.
Kane got his second of the game on the hour mark, slotting home Eric Dier's cross beyond Pickford.
West Brom, who kicked off level on points with Swansea, needed just four minutes to take the lead at The Hawthorns when Jonny Evans headed in from a corner.
Son's goal was also his 40th since joining Spurs in 2015, 24 of which have come in home games (at White Hart Lane or Wembley).
Cenk Tosun had a forgettable debut for Everton but if he is worth £27million - after scoring goals in Turkey - you have to wonder how much Son is worth. "It's up there as one of my best goals".
Everton tried to find a way back into the match and, with five minutes remaining until the break, Tosun got down the left but produced a disappointing cutback with colleagues well placed.
While Kane and Son will get the plaudits and attention for the win over Everton, Mousa Dembele was pivotal to Spurs' efforts in controlling the midfield.
His partnership with Victor Wanyama was key in the 2015/16 season that saw Spurs go close to winning the Premier League and after injury troubles for both have now passed, Pochettino can look to the duo to build another good run.
Even with that caveat in mind, few teams in the world can keep up with Spurs when they're in the kind of effervescent mood we witnessed at Wembley today.
Just one minute later and Harry Kane crowned himself Tottenham Hotspur's all time top-scorer in the Premier League.
Not even the prolific Kane has managed that feat, but the England worldwide did add another record to his growing list of accolades just two minutes into the second half when he netted his 97th Premier League goal for Tottenham, drawing level with Sheringham.