De Bruyne put City ahead from the penalty spot in the 37th minute of a dominant first half.
Newcastle did improve marginally after the break and substitute Dwight Gayle somehow missed a gaping goal.
Manchester United were the first to make their way into the semi-finals on Saturday evening, though Ole Gunnar Solskjaer's side laboured to a 2-1 win over a 10 man Norwich after extra time, captain Harry Maguire netting a 118th-minute victor for the Red Devils.
Record-holding, 13-time winners, Arsenal also made it into the last four after Dani Ceballos netted a stoppage-time victor for Mikel Arteta's side after Sheffield United's David McGoldrick had equalised following Nicolas Pepe's first-half penalty.
Newcastle were made to pay for that awful miss two minutes later when Sterling cut in from the left flank before curling one into the far corner from the edge of the box to seal the win.
Dani Ceballos' stoppage-time goal secured Arsenal's semi-final place in a 2-1 win at Sheffield United.
Chelsea midfielder Ross Barkley turned in a cross from Willian in the 63rd.
The home side carved out another opening six minutes later when Ben Chilwell's cross found Johnny Evans, who headed wide from close range.
On Sunday, Chelsea advanced to the last four with a 1-0 win at Leicester, Arsenal beat Sheffield United 2-1 and Cup holder Man City defeated Newcastle 2-0.
Mikel Arteta's charges were struggling to deal with long balls whipped inside the danger area by the hosts but they got a lifeline midway through the first stanza after they were awarded a penalty for a foul on Lacazette by Chris Basham just inside the box.