Allardyce's side have scored just three goals in their last six outings, leaving Allardyce with some work to do. Rooney saw a header disallowed for a narrow offside and then Kane, running through on the inside-left channel, drew a sharp parry from Everton goalkeeper Jordan Pickford.
The woodwork denied Son a second as he struck the left-hand post, but Tottenham were soon celebrating again as Kane surpassed Sheringham by meeting Eric Dier's curling delivery with a scruffy finish into the ground that looped over Pickford. Jonjoe Kenny completely whiffed trying to close down Son on Spurs' second, and Mason Holgate totally lost Harry Kane on the third.
The England striker tapped home two second-half efforts to surpass Teddy Sheringham's club record that had stood at 97, and take his tally to 20 league goals for the fourth season in a row.
After being called in to put out the fire caused by predecessor Ronald Koeman, Sam Allardyce has lifted Everton out of trouble, though his pragmatic style has won few friends at Goodison. It's a complete lack of discipline and a lack of fight to get back into the game.
In Cuco Martina's case some are also not good enough - yet again the Curaçao global was bad, no more so than when he backed away from Serge Aurier, allowing the Frenchman to easily deliver the cross from which Son opened the scoring on 26 minutes. I've got to cut it out, starting tomorrow.
Tottenham took the lead four minutes later with Son tapping Serge Aurier's cross into the net. If he gets a good contract and he's looked after I don't seem him chasing big dreams elsewhere.
"For me it is normal situation because I don't think anyone expected Daniel Levy to say we are going to sell".