The Rebels, who are looking to improve on the wooden spoon in their debut season, delivered a patchy performance in their first outing after a first-round bye.
Too often they let themselves down with ill-discipline and costly errors under pressure from the Waratahs, who starved the home side of quality ball.
They weren't able to string enough phases together to threaten their opponents.
Star Rebels recruit James O'Connor was one of the players guilty of allowing Carter to skip through for his first try in the 11th minute but he otherwise had a solid match with little opportunity to use his attacking flair.
Just minutes into the second half young NSW fullback Bernard Foley was rewarded with a try to give his side the bonus point in an excellent performance in his first Super start.
Melbourne made the unusual move of kicking two penalties in the second half rather than going for a try which didn't please some of their fans but it did seem to give the side some confidence as they bridged the gap.
They were then awarded a penalty try by referee Bryce Lawrence in the 64th minute and NSW hooker Tatafu Polota-Nau was given a yellow card for deliberate offside and collapsing the maul as the Rebels forwards rolled toward the tryline.
But in his absence NSW five-eighth Daniel Halangahu added a further three points with a successful penalty kick.
Lawrence also advised Rebels skipper Gareth Delve that he had given the side a white card for the citing commissioner to investigate after Carter complained of an eye gouge midway through the first half.
NSW WARATAHS 35 (Tom Carter 2, Bernard Foley, Sekope Kepu tries Daniel Halangahu 3 cons Halangahu 2, Foley pens) bt MELBOURNE REBELS 19 (1 penalty try) (James O'Connor con 4 pens) at AAMI Park. Referee: Bryce Lawrence. Crowd: 16,491.
