The Turkish outfit needed a win and for Neptunas to lose to Valencia, and that is exactly what happened - the Spanish outfit delivering a 103-65 triumph to assist Galatasaray into the next phase.
Carlos Arroyo top scored for Galatasaray with 24 points, while Patric Young had a double-double of 13 points and 13 rebounds.
Olympiacos' Matt Lajeski held his held up high in defeat, posting 22 points.
Galatasaray coach Ergin Ataman described their triumph as a "miracle", as they look to further their appearance in the last eight in 2013-14.
"I am very happy, of course. It is a historical and miracle result for us with this roster," Ataman said.
"So now, I think that our board and president will help us to build a team again for the Top 16.
"We will try to be as successful as last year and make it to the playoffs.
"Congratulations to all my players, they had a great effort tonight.
"This is a big club, one of the best in Europe.
"I think that our fans will help the club make an effort and we will get two more players for the Top 16.
"We will fight against everybody."
An even team performance saw Valencia finish Group D with three wins, not enough to see them avoid bottom - but they had a say on the Top 16's make-up with a big win over Neptunas.
No less than seven players posted double-figure points tallies, led by Luke Harangody (20) and Bojan Dubljevic (19).
CSKA Moscow finished the group phase with a perfect 10-0 record, after downing Limoges 88-56.
CSKA topped Group B, while Real Madrid (8-2) were first in Group A after beating second-placed Anadolu Efes 90-70.
Ioannis Bourousis (19 points, 10 rebounds) was impressive for Real.
Maccabi Electra were second behind CSKA, courtesy of an 81-73 victory over Unicaja.
Olimpia Milano rounded out their group campaign with a 101-96 triumph over Turow Zgorzelec, the latter finishing bottom of Group C with just one win.