The Belgian champions earned a deserved victory that booked a place in the last 32 of the Europa League, with Mbemba netting in either half against a visiting side who ended the game with 10 men after Selcuk Inan's late dismissal.
Arsenal's win over group leaders Borussia Dortmund scuppered hopes of an extended run in the Champions League, but Besnik Hasi's side still have European football to look forward to.
In truth, the score should have been more emphatic, with Aleksandar Mitrovic completely missing his kick when clean through and striking both posts with a flick.
The visitors, who have now suffered six consecutive Champions League away defeats, offered little, though Burak Yilmaz did clip the crossbar in the second half.
Their fourth defeat in five group games will heap the pressure on coach Cesare Prandelli.
Anderlecht had Steven Defour available and the hero at Arsenal, Mitrovic, was rewarded with a start.
Prandelli was without the suspended Blerim Dzemaili, and made six changes from the weekend defeat to Trabzonspor, including a recall for Wesley Sneijder.
Mitrovic missed a glorious chance in the 10th minute when he was in the clear 12 yards out to meet Frank Acheampong's low cross, but he fluffed his lines and failed to make contact.
Sneijder fired a warning to the hosts with a superb 33rd-minute free-kick that whistled just wide, but Galatasaray had an even more miraculous escape a minute later when Defour's volley was diverted by Mitrovic and hit both posts before Fernando Muslera gathered.
Anderlecht got their rewards a minute before the break when Acheampong's inswinging cross was flicked on by Dennis Praet into the path of the unmarked Mbemba to head home from 12 yards.
The visitors came out for the restart with a point to prove and Inan curled a free-kick around the wall after 53 minutes that forced Silvio Proto to punch away.
At the other end, a ricochet saw the ball find the feet of Praet who caught the shot sweetly but could only direct it straight at Muslera.
Galatasaray showed further signs of life as Yilmaz forced Proto to save with his feet, and the striker beat the offside trap 13 minutes from time but struck the bar.
Inan was sent off seven minutes from time for dissent and Anderlecht rounded off the match in style as substitute Sacha Kljestan's free-kick was parried into the path of Mbemba to tap home.