Mbemba pounced twice in both halves to beat the Turkish giants, who had Selcuk Inan sent off late on, and they now bow out of Europe completely.
Here, Sports Mole takes a closer look at what proved a bittersweet night at the Constant Vanden Stock Stadium.
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On target: 9
On target: 3
Was the result fair?
In the end it's hard to argue that Anderlecht did not deserve the three points. They were by far the better team in the first half, and weathered storms of Galatasaray pressure in the second 45 minutes before adding a killer goal in the final five minutes - after Selcuk Inan was sent off. The Turks were comfortably the better side in the second half but they ultimately did not do enough to leave Belgium with anything.
Besnik Hasi's side dominated the opening half without really threatening an opening goal. Aleksandar Mitrovic did spurn a golden chance by missing the ball at the decisive moment as he looked to fire past Fernando Muslera, but it was perhaps their only goalscoring opportunity in a first half which should have spawned far more chances for the hosts.
They only led by one goal, a lead that looked far more tenuous five minutes after the restart as Galatasaray gradually grew into the clash and probably should have scored at least once. The Belgian champions rode their luck, though, and could breathe a little easier after the Turkish side went down to 10 men following Inan's red card late on, but they made sure of the points with a second five minutes from the end.
Anderlecht drop into the Europa League now and could be a force in that competition.
The visitors produced a woeful first-half display with a performance that showed just why coach Cesare Prandelli is thought to be on his last legs as Galatasaray boss. His team showed little to no fight in the opening 45 minutes and were slightly fortunate to trail by just one goal at the break.
Did things improve after the restart? Yes, to their credit. Tarik Camdal forced Silvio Proto into his first meaningful save with a thumping free kick, while Burak Yilmaz was inches away from turning home Wesley Sneijder's cross-come-shot. Burak later hit the crossbar and it was difficult not to sympathise with the visitors.
Inan's red card on 83 minutes took the wind out of their sails, before Mbemba added a second soon after to compound a miserable night that saw them bow out of Europe completely. The clock is still ticking for Prandelli.
Sports Mole's man of the match
Chancel Mbemba: The big defender was influential at both ends; scoring the two all-important goals that secured the win and Anderlecht's Europa League participation in the New Year, but his goalscoring might detract from what was also a very fine defensive display. Full marks.
Three weeks ago Aleksandar Mitrovic proved the hero by scoring a last-gasp equaliser in a game which saw Anderlecht recover from three goals down to draw away to Arsenal. However, the Serbian was the villain of the piece at one stage tonight after missing a golden opportunity to fire the Belgians ahead.
The chance came in the first 10 minutes as Mitrovic, after being played through by Frank Acheampong, took his eye off the ball with just Muslera to beat and produced an air-kick at the decisive moment. It was a terrible miss and one that he will want to forget.
Ivan Bebek of Croatia had a busy night, booking six players in total and issuing Inan with a second yellow, which seemed a tad harsh. Inan received his marching orders on 83 minutes for protests over an earlier foul, and Bebek took exception to his behaviour and sent him off. It was quite a decent refereeing display up until that point, but perhaps he could have been a little more lenient with Inan.
Anderlecht: The Belgian champions will look to bounce back from a shock 3-1 defeat to Sporting Charleroi when they host Jupiler League leaders Club Brugge on Sunday.
Galatasaray: Galatasaray, meanwhile, could go top of the Super Lig on Saturday evening with victory away to Gaziantepspor.