Samuel Eto'o and Gary Cahill's first-half strikes secured a 3-1 aggregate victory over the timid Turks to ensure an English side's passage through to the last eight for the first time in two years.
José Mourinho's attention will now return to the Premier League title race, with Chelsea bidding to respond to the loss at Aston Villa in the Saturday showdown with Arsenal, Gunners manager Arsene Wenger's 1,000th game in charge.
"If Chelsea gives me the chance to arrive to 1,000 matches I'll be the happiest manager in the world," said Mourinho.
Mourinho has twice won the European Cup, with Porto in 2004 and Internazionale in 2010, but fell short with Chelsea during his memorable first spell.
There are numerous possible mouth-watering ties, against Barcelona, Real Madrid, Pep Guardiola's Bayern Munich, Paris Saint-Germain and Atlético Madrid, while Mourinho has not yet ruled Manchester United out of contention as David Moyes' men bid to overturn a 2-0 first-leg deficit to Olympiacos at Old Trafford on Wednesday night.
"We wait for the draw. It doesn't matter who," Mourinho said. "All the big candidates to win the competition. Welcome to any one of them."
The Ivory Coast striker, scorer of 157 goals in 341 appearances for Chelsea, scored the decisive penalty in the shootout with Bayern Munich in 2012 and was presented with a silver boot by the Blues prior to kick-off. Psychologically, Chelsea were spot-on.
"The most difficult thing was the way his team played," Mourinho said. "When you are a striker, if your team doesn't produce attacking football you feel a lonely man.
"He couldn't do much. Gary and John (Terry) had a fantastic performance. We controlled the game so well that we could never see Didier in dangerous positions."
Gala boss Roberto Mancini was scathing of his side's performance.
The former Manchester City manager said: "We did nothing on the pitch. We played really bad."