Villa boss Dean Smith has won five of his first eight games in charge, including a five match unbeaten run, to put his new team within three points of the Championship's play-off zone.
Middlesbrough, who now trail second-placed Leeds by four points, could not find a way back after James Chester had put Aston Villa ahead in the 20th minute.
Abraham, on loan from Chelsea, hit his 11th goal of a productive season to seal the points in the north east just after the hour, before substitute Glenn Whelan added the third seven minutes from the end.
After the victories for Leeds and Norwich earlier in the day, Middlesbrough and Aston Villa both had victory in mind to close the gap on the top two.
Middlesbrough were wary of the attacking threat of the visitors following their midweek 5-5 draw with Nottingham Forest.
And after Jordan Hugill had an early run at the visiting defence blocked, when he got the ball stuck under his feet, that paved the way for some pressure in the Middlesbrough half.
The danger signs had been there and how Villa did not go ahead in the 11th minute was only down to the excellence of Darren Randolph, whose afternoon soon took a turn for the worse.
The Middlesbrough goalkeeper somehow got down to his left on his line to palm away a low Abraham strike after he met Yannick Bolasie's low delivery first time.
That was a sign of things to come. Villa looked dangerous every time they broke and the home side could not get hold of the play and Abraham had a strong penalty shout dismissed when he was tripped by George Friend.
It was no surprise when Chester forced in the opener soon after. Jack Grealish's corner was misjudged by Randolph. Defenders Danny Batth and Dael Fry both lost Chester, who was able to force the ball over the line from six yards.
As half-time approached Middlesbrough went close to levelling. Mo Besic had an effort saved and Stewart Downing went close twice, once when he was unmarked at the back post and secondly his volley was blocked by team-mate Hugill.
Middlesbrough pressed after the restart but struggled to test goalkeeper Orjan Nyland. All they could muster before the hour was a Downing effort that flew high into the North Stand.
And soon after Middlesbrough manager Tony Pulis tinkered with his forward line, the Villans made it two.
Bolasie, wanted by the Teessiders in the summer, was left with space to drill a low cross into the area for Abraham to slide in and apply the finish.
And then Whelan, moments after coming on, wrapped it up when Randolph helped his long distance drive in off the left hand post.