It took the hosts just five minutes to open the scoring, as former Leicester City winger Yunus Akgun's emphatic effort nestled in the top corner, just out of Fraser Forster's reach in the Spurs goal.
Will Lankshear's first Tottenham goal brought Ange Postecoglou's side level shortly after, but the visitor's backline fell to pieces late in the first half.
Victor Osimhen netted twice in eight first-half minutes as Gala raced into a 3-1 lead, and had endless opportunities to extend that advantage after the interval.
An absolutely astonishing opening to the second half saw Spurs swamped constantly, conceding possession time and time again on the edge of the box, as Gala had seven or eight gilt-edged chances in a 15-minute period, but somehow failed to convert any of them.
Lankshear's night ended on a sour note as two quick bookings for the young striker saw the away side reduced to 10 men, but complacency set in for Okan Buruk's side as Dominic Solanke came off the bench to pull one back for the visitors, and made it a nervy final 20 minutes.
More chances came Galatasaray's way, but they looked a level below compared to how they played in the opening hour, and limped over the line in the end to win 3-2.
The win takes Gala top of the Europa League standings on goal difference with the 8pm matches still to be played, while Spurs have dropped to fifth following their first defeat on the continent this season.
GALATASARAY VS. TOTTENHAM HIGHLIGHTS
Akgun goal vs. Tottenham (6th min, Galatasaray 1-0 Tottenham)
The Ali Sami Yen erupts! Akgun recovers possession for Galatasaray on the edge of the area, and his audacious effort flashes past Forster, who got fingers to it, but was incapable of keeping the strike out.
Lankshear goal vs. Galatasaray (18th min, Galatasaray 1-1 Tottenham)
A moment Lankshear will never forget! Fine build-up play involving Son Heung-min and Archie Gray sees the latter float a pass in behind for Brennan Johnson, and the winger squares to give the young striker the simplest of finishes to equalise for Tottenham.
Osimhen goal vs. Tottenham (31st min, Galatasaray 2-1 Tottenham)
It is another Europa League nightmare for Radu Dragusin, who is caught in possession by Akgun, before Dries Mertens threads a ball through for Osimhen, and his poked finish finds the bottom corner beyond Forster.
Osimhen goal vs. Tottenham (39th min, Galatasaray 3-1 Tottenham)
Osimhen at the double! Spurs ignore the earlier warning signs that saw the Gala striker ghost into acres of space before being denied by Forster, as he latches onto the end of Mertens's superb delivery to extend the home side's advantage.
Lankshear red card vs. Galatasaray (60th min, Galatasaray 3-1 Tottenham)
On what looked to be a memorable night for young Lankshear, it has turned to disaster as the 19-year-old is dismissed for a second yellow card after a rash challenge on Gabriel Sara.
Solanke goal vs. Galatasaray (69th min, Galatasaray 3-2 Tottenham)
A remarkable turn of events sees substitute Solanke make an instant impact, only three minutes after coming on, as the Spurs striker shows good feet to deftly flick home beyond Fernando Muslera from Pedro Porro's cross
MAN OF THE MATCH - DRIES MERTENS
Osimhen will make the headlines for his brilliance brace, but Mertens was the man who assisted both of those goals, with the cross for Galatasaray's third goal being the pass of the night by some distance, and he continued to be a creative threat throughout as Gala bombarded the Spurs goal.
GALATASARAY VS. TOTTENHAM MATCH STATS
Possession: Galatasaray 54% - 46% Tottenham
Shots: Galatasaray 28 - 5 Tottenham
Shots on target: Galatasaray 10 - 3 Tottenham
Corners: Galatasaray 8 - Tottenham
Fouls: Galatasaray 14 - 10 Tottenham
BEST STATS
WHAT NEXT?
Spurs will look to maintain their fine recent domestic form when they welcome Ipswich Town to North London on Sunday, while Galatasaray face Super Lig overachievers Samsunspor here, in a meeting of Turkey's top two so far.