Coventry are now the only side in the Championship with a 100% record and Coleman could not have been happier after the game.
"Barnsley have been playing very well so we knew it was going to be hard and I thought we thoroughly deserved the three points," he said.
"Leon practises free-kicks every day in training and he despatched it in the best way."
Coventry also won their opening day game against Ipswich, but Coleman is refusing to get carried away with their impressive start.
He said: "There are another 44 games to go so there's a long way to go.
"We've got a better feeling in the squad than last year, but we still need to improve and I'm hoping to bring one or two in."
Barnsley manager Simon Davey felt that Rob Kozluk's sending off was a pivotal moment in the match, although he had no complaints about the decision.
"It was a sending off because they were two bad tackles," said Davey. "He should be more professional than that because it cost us the game.
"They were probably the only decisions the referee got right all day. He gave them a free-kick for no reason which they scored from and there were a lot of decisions he got wrong."
Davey was also critical of his team's performance. "We didn't play well and didn't create enough chances," he said.
"The performance wasn't acceptable and a few players have to look over their shoulders."
Best gave Coventry an eighth minute lead, hammering the ball past Luke Steele after Morrison won a free-kick in a central position.
The Sky Blues continued to press after the break and the task for Barnsley became harder when Kozluk was dismissed just before the hour after picking up a second yellow card.
After Best had a low shot saved and McIndoe fired over, Morrison (82) sealed the win, meeting Osbourne's cross at the far post for his third goal in two league games.