Germany winger Sane tapped home after 96 seconds and David Silva netted his 50th Premier League goal midway through the first half.
Raheem Sterling helped himself to a third goal this season as City made a similarly rapid start to the second period, Fulham's cavalier attacking approach leaving its excellent goalkeeper Marcus Bettinelli repeatedly exposed.
Pep Guardiola's champion side remains in touch with early pacesetters Liverpool and Chelsea, two points off the Premier League summit in third.
Fernandinho pounced on a dreadful pass from Jean Michael Seri and drove forward, greeting Sane's return by gifting him a simple finish.
Bernardo Silva dazzled with sublime close control before playing in Sterling, a combination of Bettinelli and his crossbar denying the England winger.
Sterling and Bernardo Silva were both involved when David Silva crashed via the underside of the bar in the 21st minute.
Ederson beat away a couple of Andre Schurrle strikes as Fulham retained their attacking intent, but the visitors were indebted to stunning Bettinelli stops from Sterling and Sane before half-time.
Sergio Aguero crossed as Sterling crowned a lightning break 98 seconds into the second half and the Argentina striker soon made way for Gabriel Jesus, who almost scored with his first touch.
There was more wayward finishing from the Brazil striker as City's forwards queued up to get in on the act, with Bernardo Silva blazing over in the 81st minute to miss out on the goal his wonderful efforts merited.
City was able to wind down in preparation for its opening UEFA Champions League match at home to Lyon, with a more combative encounter no doubt in prospect at Cardiff City next weekend.
Fulham must respond at home to in-form Watford.