Manchester City's Belgian midfielder Kevin De Bruyne (right) shoots to score the opening goal from a penalty spot
City's domination of the first half was absolute and the only surprise was it took a penalty for them to lead at halftime, De Bruyne slotting home in the 37th minute.
Newcastle did improve marginally after the break and substitute Dwight Gayle somehow missed a gaping goal.
One minute later Sterling curled home a delightful effort to seal City's progress and set up a semifinal against Arsenal.
All four of this year's Cup quarter-finals, all played behind closed doors, were won by the away team - the first time that has happened since 1987.
Manchester United will face Chelsea in the other semifinal.
Manchester United v Chelsea
Arsenal v Manchester City
Matches will be played at Wembley on July 18-19