The visitors appeared on course to make it six wins in seven after quality strikes from the Brazil international, only for the Toffees' 21-year-old centre-back, one of their players of the season, to pop up at the far post with his first goal of the campaign.
A point was as much a reward for Sean Dyche's side's determination as it was a punishment for Spurs not finishing off their opponents when they had the chance.
Richarlison chose not to celebrate his eighth and ninth goals in the last eight league matches in deference to fans who less than two years ago adored him for the part he played in their first escape against relegation.
But there were wild scenes in the fourth minute of additional time when Tottenham defender Cristian Romero could only flick on James Garner's inswinging free-kick and Branthwaite ghosted in to nod past Guglielmo Vicario, who endured a difficult afternoon dealing with Everton's set-pieces.
Dominic Calvert-Lewin's struggles also continued as he thought he had ended a 17-game drought extending back to October by nodding in from close range for the Toffees' first equaliser only for the goal to be credited to Jack Harrison just before the start of the second half.
There were no such doubts about Richarlison's well-taken goals; the first after four minutes when he swept home a volley after Idrissa Gana Gueye - injuries meaning he went came straight back in the side following Senegal's African Nations Cup exit - had allowed Destiny Udogie to run beyond him to collect Timo Werner's pass.
But instead of pressing home their early advantage Spurs were pushed back, with Harrison's embarrassment at missing from a couple of yards spared by an offside flag.
Everton's best opportunities came from set-pieces, particularly with Dwight McNeil targeting Vicario with inswinging corners from the right.
It was one of those which eventually paid off as Vicario, under pressure from Garner, failed to deal with a corner under his own crossbar and James Tarkowski headed back across goal for Calvert-Lewin to nod in.
Or so he thought until intervention from the Premier League's goal accreditation panel.
Blissfully unaware the Everton striker now had a spring in his step and only just failed to get on the end of a Harrison cross having been crowded out by centre-backs Cristian Romero and Micky van der Ven.
Everton were actually in the ascendency when Richarlison put Spurs ahead again; James Maddison and Werner combined down the left with the latter teeing up the Brazilian who, with Harrison slow to react, had time to curl a shot across and beyond Jordan Pickford.
Vicario's legs blocked Ben Godfrey's header and Vitalii Mykolenko's volley was parried late in the half but Tottenham's momentum continued after the break with Werner denied by Pickford in a one-on-one only to be flagged offside.
Pickford made a better save when it mattered to deny Richarlison his hat-trick and the visitors should have put the game beyond doubt.
That they did not should have seen Youssef Chermiti make them pay only for him to stab a shot straight at Vicario and, after fellow substitute Beto had two penalty claims turned down, up stepped Branthwaite to deliver the painful blow.
Everton are still without a league win since mid-December but the manner in which they secured a point should do wonders for morale. However, a trip to Manchester City now awaits.