Harry Kane's penalty looked set to take Tottenham third in the table after Abdoulaye Doucoure had been sent off for striking the Spurs striker in the face.
But the numbers were levelled up when Lucas Moura also saw red two minutes from time before Keane's thunderbolt salvaged a vital point in Everton's bid to beat the drop.
A draw is enough to take Sean Dyche's men out of the relegation zone and up to 15th.
Tottenham move level on points with Newcastle and Manchester United in fourth, but have played two games more than their rivals for a place in next season's Champions League.
"After the red card they were much more aggressive than us," said Stellini after Spurs failed to win away from home for the sixth consecutive game.
"We have to be a team that plays at home and away with the same desire and aggressiveness."
Stellini was taking charge for the first time since the sacking of his former boss Antonio Conte.
Conte's fate was sealed by a tirade towards the Tottenham squad after conceding twice late on to draw 3-3 at bottom-of-the-table Southampton.
But there was little sign of a fresh start as the visitors again threw away a promising position against a side fighting relegation.
Everton were lucky to escape as Kane missed two huge chances inside the first 15 minutes.
The England captain should have been full of confidence after becoming his country's all-time top goalscorer during the international break.
But Kane's scuffed effort gave Keane the chance to clear off the line with his first opportunity before he headed wide when completely unmarked from Ivan Perisic's cross.
No side has scored fewer goals in the Premier League than Everton this season and the home side's lack of a focal point up front without the injured Dominic Calvert-Lewin was again exposed.
The home outfit did waste a glorious chance to take the lead when Idrissa Gueye selfishly blazed over from the edge of the box with teammates better placed to take the shot just after half-time.
A moment of madness from Doucoure appeared to have undone the endeavour of Dyche's men for the first hour.
The French midfielder slapped Kane in the face after an altercation between the pair and could have no complaints at being shown a straight red card.
Everton's self-destruction did not stop there as moments later Keane dived in foolishly on Cristian Romero to concede a penalty.
Kane coolly sent international teammate Jordan Pickford the wrong way for his 24th club goal of the campaign.
"The sending off changes the feel, but our mentality is growing all the time," said Dyche.
"They're pushing their own standards very high, even going down 11 v 10, we took the game on," he added.
Tottenham failed to make the man advantage count for much of the closing stages as they invited Everton on in search of an equaliser even before Lucas also saw red for a rash challenge on Keane.
Everton's fight got its reward when centre-back Keane strolled forward before unleashing a thunderous hit into the top corner.
