After a low-key start, Coventry should have taken the lead in the 12th minute when Michael Mifsud outpaced Barnsley skipper Paul Reid inside the box only to watch his shot hit the left and then right post.
The visitors responded with an excellent chance when Sam Togwell's drive from 20 yards clipped the bar.
Mifsud had two further chances when his close-range shot from Adebola's cross was blocked by right-back Neil Austin and he then again outpaced Reid only to fire a shot high into the crowd.
The hosts finally made their superiority count when they took the lead in the 27th minute.
Colin Hawkins knocked the ball into the path of Jay Tabb and his powerful 20-yard shot ended up in the roof of the net despite keeper Nick Colgan's best efforts.
Coventry extended their lead five minutes later when Stephen Hughes teed up Marcus Hall and the left-back's centre was driven home from six yards by Mifsud.
Barnsley conceded a third goal in the 38th minute when Isaac Osbourne found Mifsud in space on the right and his cross was driven high into the net by forward partner Adebola.
Coventry didn't sit back after the break and notched a fourth goal five minutes after the break.
Adebola raced on to a long skied clearance from Hawkins and out-muscled Reid before firing home his ninth goal of the season with a ten-yard strike.
Simon Davey's men finally beat Coventry's defence in the 79th minute when Coventry-born Daniel Nardiello's corner from the left was headed over the line by Bobby Hassell.