Giannakopoulos scored with his first touch of the ball
Bolton held on for a valuable point in their battle against relegation despite being totally outplayed by Tottenham.
Spurs dominated the first half, but failed to create clear chances against a well-organised Bolton defence.
The visitors grabbed a shock lead when substitute Stelios Giannakopoulos stabbed home a scruffy goal from close range within 20 seconds of the restart.
Steed Malbranque turned in Robbie Keane's shot to level before Dimitar Berbatov wasted a glorious late chance.
With new signing Luka Modric watching from the stands, Spurs had more possession than they knew what to do with in a one-sided first half.
Gary Megson believes his side proved they can escape relegation
For all their dominance of the ball, Spurs could have done with a little more of Modric's play-making ability on the pitch as they flattered to deceive.
Robbie Keane's optimistic effort with the outside of his right boot was comfortably saved by Ali Al Habsi, while Jermaine Jenas chipped in a delicate ball over Berbatov's shoulder which went just too far for the striker.
Keane and Malbranque both saw curling efforts end up wide of the target, but Spurs did not make Al Habsi work hard in the Bolton goal.
Bolton offered next to no attacking threat with their tactic of lumping long balls up to Grzegorz Rasiak failing to trouble the Spurs defence at all.
For all their toothlessness in attack, Bolton were defending resolutely with a well-organised formation designed to frustrate Spurs.
In the dying moments of the first half, Spurs came agonisingly close to taking the lead when Gilberto's dangerous free-kick was diverted onto his own post by Gretar Steinsson.
The ball bounced away for a corner, but Bolton did enough to hold on until the break.
The second half got off to a bizarre start with Spurs full-back Alan Hutton, who had run out to take his place, called over to the other side of the pitch to be replaced by Tom Huddlestone.
Spurs' concentration at the back certainly seemed to be lacking moments later as Stelios - with his first touch of the match - pounced on a loose ball in the six-yard box to stab his side into the unlikeliest of leads, just 19 seconds after the restart.
Poyet was frustrated by the lack of finishing from his side
Bolton's goal shocked Spurs into a more urgent approach, and they deservedly equalised within six minutes.
Keane's ball to the far post found an unmarked Malbranque racing in to slide in past Al Habsi.
Spurs began to play some lovely attacking football, with Berbatov, Keane and Malbranque linking up to create a series of opportunities for their side.
Aaron Lennon had a fierce shot well saved by Al Habsi, Gary Cahill made a superb last-ditch challenge on Keane, and Danny Guthrie cleared Darren Bent's header off the line.
And deep in injury time, an unmarked Berbatov skied his header from Jamie O'Hara's free-kick into the box.
The result means Bolton have taken seven points from the last nine available as they stay out of the relegation zone.
Tottenham first team coach Gus Poyet:
"I'm a little bit disappointed - we played good football and we were good on the ball and created chances but we didn't score.
"It's up to the players to finish. That's why the most expensive players are the ones who score goals.
"We are not able to score the second goal and that is a problem."
Bolton boss Gary Megson:
"We've battled manfully although I accept we didn't play very well.
"But in the situation that we're in, coming down to Tottenham and getting a point is a good result for us.
"I'm sure there are people who looked at us two weeks ago and thought we'd gone but we haven't."
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