O'Brien (19) got on the end of a long ball from Jean-Yves M'voto and then lobbed goalkeeper Joe Murphy to put Barnsley in front.
Barnsley's Tomasz Cywka then twice went close to putting the home side further ahead, testing Murphy with a 30-yard free-kick and hit the keeper's left-hand post with a curling effort.
Moussa (78) equalised after bursting through the middle before beating Luke Steele to set up a stirring finish.
Clarke struck the winner just a minute from time after holding off a couple of defenders.
Pressley said: "Both goals were outstanding finishes and I'm delighted for everybody connected with the club.
"I thought that Leon Clarke in particular was outstanding, leading the line.
"The most important thing for us was that we started believing in ourselves.
"In the opening 30 minutes, we never managed to pass the ball in the manner that people have come accustomed to with a Coventry side.
"I thought we were terrific in the final 15 minutes of the first half and in the second half.
"We controlled the game and passed the ball with much greater speed. Like normal, we pressured the game brilliantly.
"I think the one thing that's very important is that we've been behind in the last four games and we've gone on to win three of them and draw the other.
"That shows the type of spirit and mentality within the group at this moment in time.
"I am extremely proud of the way they're going about things. They will now have three games off because they've worked exceptionally hard over the Christmas period."
- Sports & Recreation
- Leon Clarke
- Franck Moussa
- Joe Murphy