SPURS tossed away a two-goal lead as they came Acropolis in Athens.
Harry Kane fired the North Londoners ahead against the run of play with a 26th minute penalty.
Last season’s semi-final hat-trick hero Lucas Moura added another four minutes later with a sizzling shot from outside of the area as Tottenham looked in total control.
But Daniel Podence pulled a goal back for Olympiakos in the 44th minute.
And after VAR had confirmed the assistant ref was right to rule out a Dele Alii ‘goal’ for offside, the Mathieu Valbuena made it 2-2 with the second spot kick of the night after he had been fouled by Vertonghen.
Alli and sub Erik Lamela both had shots saved by hosts’ keeper Jose Sa late on as Spurs went in search of a winner.
And while Olympiakos are now unbeaten in 19 matches in all competitions since February, Tottenham boss Mauricio Pochettino and his players will be disappointed to have thrown away a two-goal lead just like they did at Arsenal earlier this month.
Pochettino made five changes to the starting line-up from Saturday’s 4-0 win at home to Crystal Palace, with seven of those who started in Madrid keeping their places.
They included Christian Eriksen, who arguably had one of his poorest performances in a Spurs shot.
Kane’s strike was his 15th Champions League goal in just 20 appearances.
It may not have been enough to have earned his team a win but with one point they at least have the same number as they did from their opening three group matches last season.