The win was Villa’s second of the season in eight matches and lifted the club to 12th place from having arrived at Oakwell Stadium place 18th.
Adomah seems to have managed to out behind his early struggles that have seen him continuously overlooked by both Black Stars and Steve Bruce.
The 29-year-old opened the scoring with a close-range effort in the 19th minute before doubling the visitors lead with a 44th-minute penalty.
Kienan Davies completed the Villans’ the rout with a 55th-minute strike.
Villa had gone three matches without a win prior to this encounter and the win eases the pressure on Bruce who singled out the former Middlesbrough for praise after the match.
Even though Adomah had scored in Villa's 4-1 thumping of Wigan Athletic in the League Cup recently, the strikes were his first in four Championship outings this campaign.
Meanwhile, Barnsley continued to overlook Andy Yiadom after his botched move to Huddersfield Town in the offseason.