Earlier, Premier League leaders Arsenal stayed on track for a historic quadruple by seeing off a feisty challenge from third-tier Mansfield, to win their fifth-round tie 2-1.
City coach Pep Guardiola, juggling punishing Premier League and Champions League commitments, changed nearly the entire side from the 2-2 draw with Nottingham Forest in midweek, with only Matheus Nunes keeping his place.
Eddie Howe made a relatively modest four changes to his Newcastle team, fresh from their late win against Manchester United in midweek, with match-winner Will Osula given a rare start.
The home side began strongly at St James' Park and took a deserved lead when Harvey Barnes finished smartly after being found by Sandro Tonali.
City were level in the 39th minute in bizarre fashion when Jeremy Doku's dangerous ball behind the defence ran all the way to Savinho.
The Brazilian stood still and appeared to let the ball bounce off his leg into the net.
Soon after the restart, City completed the turnaround when Omar Marmoush smashed the ball into the roof of the goal from close range.
The Egyptian forward then thundered home his second in the 65th minute to put his team in total control.
Guardiola turned to a bench full of international talent to refresh the team, who suffered few scares in reaching their ninth FA Cup quarter-final in the past 10 years.
City, who travel to face Real Madrid in midweek, remain in the Premier League title hunt but are seven points behind Arsenal after their stumble against Forest.
