Last month boss Unai Emery dismissed talk of his side pushing to become English champions, instead saying they were unlikely to finish in the top five - and this game against Leeds was another sign they are not ready to challenge in the title race.
The result means they have only picked up five points from their past four matches, although they remain in a strong position to qualify for next season's Champions League.
Before the match, writing in the matchday programme, Emery was trying to remind the fans the club were having a fine season despite recent results going against them.
He wrote: "We are at a beautiful and fantastic moment to compete in two competitions for the remainder of the season. It is the moment to show our wishes and follow our dreams.
"We have a great opportunity to achieve our objectives and dream of big things. Our position in the table is enviable and we all would have signed up for it before the season began."
Nevertheless, this was a chance missed against a side with such a poor away record and there was a smattering of boos from the home fans at half-time.
But those boos turned into cheers with the late goal.
Villa dominated the second half and, after the introduction of Jadon Sancho and Abraham, they piled on the pressure.
Abraham, who scored 25 goals in 37 Championship appearances for Villa during a loan spell from Chelsea in 2018-19, was the one to make the breakthrough.
He had grabbed a goal in the 3-1 loss to Newcastle in the FA Cup a week ago and he was on target again to the delight of the Villa faithful.