The Turkish champions have 10 points from four games and have all but secured qualification for at least the knock-out play-off round, while Spurs sit in fifth place after losing their 100-percent record.
Galatasaray took an early lead when Yunus Akgun smashed a sensational long-range volley into the top corner in the sixth minute.
But the visitors hit back just 12 minutes later as teenage striker Will Lankshear marked just his second first-team appearance with a goal by tapping in Brennan Johnson's pass across goal.
Galatasaray took control before half-time, though, as Osimhen scored twice in the space of eight minutes.
The Nigerian star fired them back in front just after the half-hour mark, before cleverly volleying Dries Mertens' cross past Tottenham goalkeeper Fraser Forster.
Osimhen has now scored six goals in eight appearances for Galatasaray since joining on loan from Napoli.
The home side passed up several opportunities to put the game out of sight for their opponents, but Spurs' hopes of a comeback were hit when Lankshear was sent off for two yellow-card offences in quick succession in the second half.
Dominic Solanke pulled one back in the 69th minute, shortly after being brought on by Ange Postecoglou, but Galatasaray held on with relative comfort.
Eintracht Frankfurt are second in the standings, behind Galatasaray on goals scored, after Omar Marmoush's 13th goal of the season clinched a 1-0 win against Slavia Prague.
The Egyptian clipped a superb free-kick in off the crossbar shortly after the break.
Athletic Bilbao are in third, separated from Eintracht also by goals scored, after coming from behind to grab a 2-1 success at Ludogorets.
Inaki Williams and Nico Serrano both scored in a two-minute burst from the Basque club.
Roma's disappointing form this season continued as Kevin Mac Allister's goal 13 minutes from time secured Union Saint-Gilloise their second point of the competition with a 1-1 draw in Belgium.
