Spurs, who were without Harry Kane due to illness, opened the scoring through Dele Alli midway through the second half but were reduced to 10 men with just over 10 minutes of normal time remaining as an injury to Andre Gomes overshadowed the game.
Son Heung-min was shown a straight red card for a challenge on the Portuguese midfielder, which left Gomes with what appeared to be a serious leg injury.
The 26-year-old fell awkwardly as Son tackled him and required extensive treatment on the pitch before he was removed on a stretcher.
A visibly distraught Son was sent off by by referee Martin Atkinson as Gomes received treatment.
Earlier, Alli had given Spurs the lead when he capitalised on some sloppy play from Alex Iwobi and fired home.
Everton arguably should have had a penalty soon after when Alli appeared to handle the ball inside the box but the spot kick was not awarded following a three-minute VAR check.
The hosts did manage to claim a share of the spoils, however, as Cenk Tosun headed home from close range deep into stoppage time with Lucas Digne providing the assist.
Spurs, who are without a league win on their travels since January, now have 13 points from 11 games and sit in 11th spot.
Everton are in 17th place and are three points clear of Southampton in the final relegation position.