City’s goals were scored by debutant Siphamandla Dondolo (45th minute), Jaedin Rhodes (53) and substitute Jody Lee Ahshene (81).
With this result, City will join neighbours Stellenbosch, Magesi FC, Marumo Gallants, Mamelodi Sundowns and Kaizer Chiefs in the quarter-final round. The quarterfinal draw will take place on Sunday.
Both teams played at a hectic pace at the start but the slippery conditions saw movements lack penetration because often there was a possession turn-over.
City could have ended the match as a contest by halftime because they had at least four gilt-edged scoring chances but Dondolo was the only one to capitalise.
Royal AM showed some delightful touches in the first half and traded heavily on a short-passing game but their sorties lacked direction and others broke down on the fringes of the opposition goalmouth.
Once City went ahead 2-0 Royal AM dropped their heads and the sting went out of their challenge.
Midway through the second half, City coach Eric Tinkler made a raft of substitutions, one of which saw the introduction of 19-year-old Ahshene, the SA Under-20 cap. He hammered the final nail in the Royal AM coffin after side-footing a goalmouth cross into the opposition net.
This first-round Carling Knockout defeat follows hard on the heels of a telling blow Royal AM suffered earlier this week. The Court of Arbitration for Sport (CAS) threw out their appeal to have their FIFA transfer ban overturned.
Royal AM will now turn their attention to a fixture against Mamelodi Sundowns in the Betway Premiership next Wednesday.
City will next play Magesi in a Premiership next Sunday.
Other results:
King Zwelithini Stadium: AmaZulu FC 1 Stellenbosch FC 2 (after extra time)
Orlando Stadium: Orlando Pirates 2 Magesi FC 3
Old Peter Mokaba Stadium: Polokwane City 2 Marumo Gallants 3
Lucas Moripe Stadium: Mamelodi Sundowns 5 Lamontville Golden Arrows 0
New Peter Mokaba Stadium: SuperSport United 0 Kaizer Chiefs 3