Last term's beaten finalists were two goals to the good after half an hour in Athens thanks to Harry Kane's penalty and Lucas Moura's excellent finish.
Mathieu Valbuena was the catalyst for a deserved Olympiacos comeback, though, setting up Daniel Podence for a goal on his debut in the competition before burying a penalty 54 minutes in.
2 - Tottenham failed to win a Champions League game having been two goals ahead for the second time and first since drawing 2-2 with Werder Bremen in September 2010. Pegged. pic.twitter.com/fMO0SnG3Fw— OptaJoe (@OptaJoe) September 18, 2019
Olympiacos seemed to target Davinson Sanchez at right-back in the early stages and one such attack led to Guerrero rattling the base of the left-hand post with a shot from just inside the area.
The home side were certainly the stronger until Kane fired a spot-kick high into Jose Sa's net, having been tripped by Yassine Meriah as he worked his way into the box.
There was another blow to Olympiacos four minutes later, as Ben Davies stole back the ball in their half and fed Lucas, who lashed a shot beyond Sa from the edge of the area.
Pedro Martins' side grabbed a lifeline before the break, though, Podence exchanging passes with Valbuena and firing past Hugo Lloris' right hand.
Spurs were then punished again when some slack play from Christian Eriksen led to Jan Vertonghen standing on Valbuena's foot, allowing the French midfielder to send Lloris the wrong way from the spot.
Dele Alli was denied by the legs of Sa, who also kept out an effort from Erik Lamela, as Spurs failed in a late push for a winner, meaning they still have not won back-to-back away matches in Europe's elite competition.
What does it mean? Spurs frailties a growing concern
The problems in Mauricio Pochettino's defence were apparent last term even as Spurs reached the final – indeed, they have now conceded 20 goals since the start of last season's competition, more than any other team.
While most would still expect them to progress from Group B, this was a stark reminder of Spurs' fallibility at the back. They might not have merited a 2-0 lead after 30 minutes, but a team of their standing should certainly not have surrendered it so meekly.
Olympiacos dance to Valbuena's tune
Valbuena set up Olympiacos' first goal and won and converted their penalty, capping a fine display in which he largely dictated the pace of the game.
Eriksen a shadow of himself
When Spurs threw away a two-goal advantage at Arsenal this month, it was a result that did not do Eriksen's excellent display justice. On this occasion, he summed up Spurs' frailties completely, offering next to nothing in attack and conceding possession in the build-up to Valbuena's penalty.
Spurs – who face Leicester City in the Premier League on Saturday – play their second group game at home to Bayern Munich on October 1, when Olympiacos head to Red Star Belgrade.