The Magpies, backed by over 6,500 away supporters this evening, conceded in the 14th minute as Abraham latched onto Douglas Luiz's free-kick inside the box before sweeping the ball beyond United goalkeeper Aaron Ramsdale.
The striker was situated in an offside position following Luiz's pass but, with no VAR in place for the cup tie, the goal stood.
Harvey Barnes, who scored a brace in Newcastle's FA Cup third round win against AFC Bournemouth, went close to drawing the visitors level as the left winger completed a neat one-two with William Osula before crashing an effort into the side-netting.
Villa were fortunate not to receive a red card prior to Marco Bizot's dismissal as Lucas Digne committed a late foul on Jacob Murphy, with referee Chris Kavanagh only brandishing a yellow card to the Villans captain.
Bizot, however, was sent off on the stroke of half-time as the Dutch goalkeeper raced off his line before taking out Jacob Murphy with a wild swipe towards the half-way line.
United booked their place in the fourth round after beating AFC Bournemouth via penalties after last month's thrilling 3-3 draw at St. James' Park.
Newcastle travelled to the second city for last season's FA Cup fourth round tie, coming from behind to claim a 3-2 win against Birmingham City.
**Newcastle United:** Aaron Ramsdale, Kieran Trippier (c), Lewis Hall, Sandro Tonali, Harvey Barnes, Malick Thiaw, William Osula, Jacob Murphy, Nick Woltemade, Dan Burn, Jacob Ramsey.
**Subs not used:** Nick Pope, John Ruddy, Anthony Gordon, Anthony Elanga, Joe Willock, Alex Murphy, Leo Shahar, Sean Neave, Seung-soo Park.