Steven Naismith scored twice in the first half for Norwich but goals from Owen Bailey, Mohammed Sangare and Macaulay Gillesphey gave the visitors a 3-2 half-time lead.
But the home side responded in the second half with Todd Cantwell equalising from the penalty spot and Abrahams sealing Norwich's third win in four matches with a goal 16 minutes from time.
Norwich move up to sixth place, three points behind fifth-placed Fulham, while Newcastle are in 10th.
See: Norwich City report
Norwich: 26 Jan v Blackburn (A) PL2
Newcastle: 18 Jan v Burnley (A) PL Cup