Coventry were handed a free-kick 20 yards out after eight minutes when striker Morrison went down easily under a tame challenge from Stephen Foster.
It brought protests from the home side that were brushed away by referee Steve Tanner and Best smashed an unstoppable free-kick from 20 yards out to beat Luke Steele before he had time to react.
Veteran striker Jon Macken dragged a shot wide and a cross from Jamal Campbell-Ryce was held at the second attempt by Keiren Westwood.
The visitors should have stretched their lead after 38 minutes but Isaac Osbourne blazed a wild shot wide after Morrison picked him out with a short pass.
Barnsley's best chance came in the dying minutes of the first half when Adam Hammill's cross from the right had the City defence back-pedalling and Stephen Wright headed just wide of his own goal.
After the break, Coventry threatened to add to their lead as Aaron Gunnarsson broke but he fired wide and Foster had to slide in front of Morrison to clear soon afterwards when Gunnarsson drilled in a low cross.
Barnsley were down to ten men after 58 minutes when full back Rob Kozluk was booked twice in as many minutes.
First he slid into Wright then produced a wild tackle on Martin Cranie that saw him red carded.
A bright spot came for Barnsley when Canada international Ian Hume came on after 66 minutes to make his his first competitive appearance since his fractured skull against Sheffield United last November.
Nine minutes form the end City grabbed the second when Osbourne crossed to the far post where unmarked Morrison had an easy tap in.