The hosts got the evening off to a dream start as Samuel Eto'o fired them in front after just four minutes, beating the offside trap to latch onto Oscar's through ball as his strike was too strong for the hand of goalkeeper Fernando Muslera.
Chelsea continued to control proceedings but had little joy in front of goal, with Eden Hazard, Willian and John Terry all spurning chances, but Gary Cahill doubled the scoreline just before the break, slamming home from close-range after Muslera pushed Terry's header into his path.
The visitors rarely threatened and Didier Drogba was largely subdued on a frustrating Chelsea return. The Blues came close to extending their lead late on as Muslera denied Hazard and then substitute Fernando Torres with smart saves on a comfortable evening for Jose Mourinho's men.
Mourinho warned of a dangerous tie poised 1-1 after the first leg in Istanbul three weeks ago but four minutes into the second leg, his side regained the advantage before advancing into the last-eight draw with ease - 3-1 on aggregate.
Drogba, scorer of 157 goals in 341 appearances over eight years for the Blues, including the decisive penalty in the 2012 European Cup win, was expected to be the main protagonist of the contest.
But it was another 30-something African striker who found the net with the first attack as, for once, the Ivorian faded into the background on the big occasion, with Hazard the outstanding forward on display.
Eto'o showed his predatory instinct to fire the Blues in front at the end of a move started by Hazard and although Chelsea were unable to add to their tally, they did not need to.
Roberto Mancini's visitors were a distant second best throughout and offered up little resistance as the former Manchester City manager's return to England ended in a limp loss.
Chelsea - and Hazard, in particular - were finding holes in the visiting defence and Felipe Melo was booked after a series of fouls, usually on his fellow Brazilians.
Hazard's run had helped to set up the goal and the Belgian's one-two with Oscar led to a chance for Lampard, which the midfielder could only prod over.
Hazard showed great strength to out-muscle former Arsenal right-back Emmanuel Eboue on the right wing and tried to fashion another chance only to be crowded out at the byline and then had a shot saved.
Frank Lampard's free-kick was volleyed narrowly over by Terry and a Drogba set-piece from a similar distance at the other end was so far over, it hit the banner bearing his name.
Hazard and Eto'o combined to set up a chance for Willian, who could only weakly direct the ball straight at Muslera, but soon Chelsea had their second.
Lampard and Terry combined at another set-piece and this time the captain's header from a corner was saved, with Cahill following up to thrash the ball into the net.
Chelsea set about trying to add to their lead and went close in the opening skirmishes of the second period.
Oscar was unable to test Muslera after a mesmerising Hazard run, before the Galatasaray goalkeeper saved from Willian and Oscar shot wide after cutting in from the left.
Lampard headed straight at Muslera as Chelsea continued to search for a third, before Drogba was booked for tugging back Cesar Azpilicueta.
Azpilicueta chipped off target before Hazard was denied by Muslera. Then Torres, who replaced Eto'o, was teed-up by Hazard and denied by the visiting goalkeeper as Galatasaray and Drogba were eliminated.