First-half goals from Samuel Eto'o and Gary Cahill gave Chelsea F.C. a 2-0 victory as they saw off their Turkish opponents 3-1 on aggregate to succeed where fellow Premier League clubs Arsenal and Manchester City F.C. failed and make it into the last eight.
Meanwhile, Carlo Ancelotti's Madrid defeated Schalke 3-1 at the Santiago Bernabeu to advance 9-2 on aggregate and extend their long unbeaten run to 31 games.
"Last season when Chelsea F.C. won the Europa League it was a different level," Chelsea manager Jose Mourinho said.
"Now we are amongst the last eight in the Champions League, the best teams in the world.
"That is important for the club. The players deserve to be there."
The build-up to the game at Stamford Bridge had been dominated by the return of Didier Drogba to the ground where he is still hailed as a hero and the Ivorian - scorer of the winning penalty when Chelsea F.C. beat Bayern Munich in a shoot-out in the 2012 final - was presented with a silver boot prior to kick-off.
Chelsea F.C.'s away-goal advantage from last month's meeting in Istanbul meant they had no obligation to take the game to Roberto Mancini's side, but they found themselves ahead on the night inside four minutes.
Oscar's pass in behind the visiting defence found Eto'o, and the Cameroonian beat the offside trap before sending in a low shot that was too powerful for goalkeeper Fernando Muslera.
As Eto'o celebrated scoring the 30th Champions League goal of his career, Drogba and Galatasaray offered little by means of a response.
John Terry volleyed just over from a Frank Lampard free-kick before Jose Mourinho's side increased their lead three minutes prior to the interval, Gary Cahill ramming home the rebound after Muslera had blocked Terry's header at a corner.