Last term's beaten finalist was 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.
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 was 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.