With new manager Roberto De Zerbi talking charge at home for the first time on Saturday (Sunday AEST), Spurs looked set to end a 15-match winless in the top flight when Xavi Simons produced a sensational 77th-minute strike.
Xavi's fine effort, which sparked wild celebrations, followed his assist for Pedro Porro's opener after 39 minutes, which was cancelled out by Kaoru Mitoma in first-half stoppage time.
But jubilation turned to disbelief when Kevin Danso lost possession in the fifth minute of time added on and Rutter curled home to ensure the spoils were shared.
It means Tottenham will stay in the bottom three for another week, with only five matches left to prevent a first relegation since 1977.
Tottenham are a point behind West Ham, who visit Crystal Palace on Monday, and two behind Nottingham Forest, who host Burnley on Sunday.
Leeds manager Daniel Farke insisted "nothing is achieved yet" after his side took another big step towards EPL safety with a 3-0 home win against all-but-relegated Wolves.
Goals from James Justin and Noah Okafor put Leeds 2-0 up inside 20 minutes, before Dominic Calvert-Lewin's stoppage-time penalty sealed back-to-back EPL wins for the first time this season.
"To be there with 39 points, with five game-days to go, is a fantastic position to be in, but the feeling is still that we probably need to win one, perhaps even a few more points," Farke said.
Manchester United opened up a 10-point advantage in the race for the Champions League after Matheus Cunha's first-half strike earned a 1-0 win over Chelsea at Stamford Bridge, which seriously harmed the Blues' top-five hopes.
Cunha rifled in the winner with United's only shot on target as home fans' anger at the running of their club reverberated around the ground, following a massive street protest against owners BlueCo that took place ahead of kick-off.
This was Chelsea's fourth Premier League defeat in a row and the fourth game in which they have failed to score.
They trail Liverpool by four points in the race for the top five having played a game more. With five teams within a win of catching them, they are now at serious risk of failing to qualify for Europe.
For Michael Carrick's third-placed United, there are no such worries. This was a welcome return to form after one victory in four matches.
Under-pressure Newcastle United manager Eddie Howe has insisted he will not waver in his belief that he is the right man for the job despite another dispiriting defeat.
The Magpies' hopes of securing European football for next season were dealt another significant blow when Bournemouth, Howe's former club, left St James' Park with a 2-1 win, condemning the hosts to an eighth defeat in 11 EPL outings and a fifth in six at home.
Having heard his team leave the field at both halftime and on the final whistle to boos, Howe was asked if had started to doubt himself.
"My belief in myself can't waver, and it's not,'' he said.
"I have to retain a real belief in my work and my staff and what I do."
