Osimhen scored his first goal in Europe for Galatasaray after his transfer from Napoli with a precise low shot inside the right post to give the hosts a 2-1 lead in the 31st minute.
He then added his second eight minutes later by heading home a curled cross from the right by Dries Mertens.
Yunus Akgün put Galatasary ahead in the sixth minute when Spurs failed to properly clear a free kick by Mertens and the midfielder volleyed home from outside the area past goalkeeper Fraser Forster.
Tottenham equalised in the 19th minute through 19-year-old striker Will Lankshear, who made only his second senior appearance for the London club and netted from close range after a cross from Brennan Johnson.
However, Lankshear was later sent off for a second yellow card on the hour mark.
Substitute Dominic Solanke pulled a goal back for Tottenham with a backheel finish in the 69th to set up a thrilling finish. but they couldn't find an equaliser.
Spurs boss Ange Postecoglou made seven changes from Sunday's win over Aston Villa with three teenagers in Archie Gray, Lucas Bergvall and Lankshear included in the starting line-up
Tottenham boss Ange Postecoglou made seven changes to the team that beat Aston Villa. (AP PHOTO)
It was Tottenham's first loss in the competition after three wins. Galatasaray remain unbeaten with three wins and a draw.
Every game on Thursday was preceded by a moment of silence to honour the victims of the devastating floods in Spain.
At Old Trafford, Amad Diallo's delightful double ended Manchester United's 380-day wait for a European win as Ruud van Nistelrooy oversaw a much-needed 2-0 victory against PAOK in his penultimate match as interim boss.
The result extended Van Nistelrooy's unbeaten interim stint to a third match, with incoming head coach Ruben Amorim watching remotely from Portugal ahead of taking charge at Old Trafford on Monday.
Diallo will have certainly caught his eye on his first start in five weeks, with the lively winger denied a penalty despite appearing to be caught by Baba Rahman in a forgettable first half.
The 22-year-old continued to prove a thorn in the side and fantastically guided home a header before settling nerves with a beautiful curling effort from distance after winning the ball.
Amad Diallo scored twice in Manchester United's first win of their Europa League campaign. (AP PHOTO)
In other matches, Eintracht Frankfurt beat Slavia Prague 1-0 thanks to Omar Marmoush's second-half goal, while Roma were held to a 1-1 draw at Union Saint-Gilloise, another disappointing result after a 3-2 loss to Hellas Verona in Serie A over the weekend.
Rangers held last year's Conference League champions Olympiacos to a 1-1 draw in Greece.
Athletic Bilbao beat Ludogorets 2-1 to stay unbeaten with three wins and a draw, while Qarabag upset Bodø/Glimt 3-1 in Norway to earn their first points.
Nice drew 2-2 with Twente with both teams finishing the game with 10 men.
AZ Alkmaar defeated Fenerbahce 3-1, Ajax thrashed M Tel Aviv 5-0 while Ferencvaros whipped Dynamo Kyiv 4-0.
Hoffenheim and Lyon drew 2-2, while Lazio edged out Porto 2-1.
In the Conference League, Legia Warsaw and Rapid Vienna both kept a perfect record in the third-tier competition by making it three wins from three games.
Warsaw routed Dinamo Minsk 4-0 and Vienna eased past Petrocub 3-0 in Moldova.
EPL giants Chelsea fielded a second-string side but still thrashed Armenian side Noah 8-0, Apoel defeated Fiorentina 2-1 while Hearts lost 2-0 at home to Heidenheim.
Djurgarden beat Panathinakos 2-1 while Vitoria SC topped Mlada Boleslav 2-1. LASK and Cercle Brugge drew 0-0.
with PA
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