David McGoldrick struck twice to help the home side continue their climb away from the foot of the table, taking his tally to 11 since joining on loan from Nottingham Forest.
John Fleck opened the scoring in the ninth minute before McGoldrick reached double figures right on the stroke of half-time. Crawley, who went into the game in third place, pulled one back in the 56th minute when Billy Clarke got the goal he deserved but McGoldrick's second settled matters.
"I thought we were excellent," said Barker. "We were by far the better side, we created the best chances, we played the best football and we are disappointed not to have won the game.
"Coventry must have gone away from this wondering how they won the game. It won't happen all the time. I was actually very proud of the way we passed the ball about.
"To come to a place like this and create as many chances as we did is very good. I have got no complaints against my players at all. In the end the difference came down to a player who should be playing at a higher level."