Barnsley woke up after the restart and hauled themselves back into the game in the 50th minute through Jon Macken.
Mark Robins' side pushed hard for an equaliser but they could not make their pressure pay and Sky Blues substitute Clinton Morrison added a late third to confirm Barnsley's first defeat in nine league matches.
Coventry boss Chris Coleman made two changes to his starting line-up, replacing Richard Wood and Morrison with Leon Barnett and Leon Best.
Robins made two changes to the side that lost 1-0 to Scunthorpe in the third round of the FA Cup last weekend.
Jacob Butterfield and Andy Gray paid the price for the Glanford Park defeat as Darren Moore and Macken were restored to the side.
Barnsley made a bright and inventive start and Stephen Foster found himself with a shooting chance after just two minutes.
The defender was afforded the time and space to fire goalwards from six yards but Sky Blues goalkeeper Keiren Westwood made a fine block to avert the danger.
Two minutes later, Foster threatened again when he rose to plant a firm header goalwards, but could only watch in agony as the ball hit the crossbar.
Coventry, after a slow start, began to stamp their authority on proceedings and engineered a decent opening in the 18th minute.
Clingan crossed dangerously and Aron Gunnarsson greeted his delivery with a thumping header from 12 yards.
Barnsley defender Carl Dickinson made a fine clearance and Leon Best could only drill the rebound wide from just six yards out.
The visitors signalled their threat again in the 22nd minute when Hugo Colace hit a brilliant volley from 20 yards and the highly-rated Westwood made an equally fine save.
However, Coventry then hit their opponents with two quickfire goals that put them in complete control.
In the 24th minute, former Liverpool defender Stephen Wright flighted the ball in from the right flank and Eastwood headed home in style from 12 yards out.
Three minutes later, their advantage was doubled as Clingan volleyed a loose ball into the bottom corner from 12 yards.
Gunnarsson then tried his luck from 30 yards but Tykes defender Bobby Hassell did well to make a fine block.
Barnsley boss Robins made two changes at half-time, replacing Anderson De Silva and Daniel Bogdanovic with Adam Hammill and Iain Hume.
The changes breathed new life into the visitors as an attacking force and they were right back in the game within five minutes of the restart when Macken headed in a cross from Hammill.
Tykes midfielder Emil Hallfredsson and Hume both threatened for the visitors before Hammill had a shot blocked by Coventry defender Martin Cranie.
And Coventry sealed the points in the second minute of added time when he headed home from 12 yards after a fine cross from Best.