The Sky Blues took the lead when Forest centre-half Luke Chambers headed into his own net, but responded well to level in fine style through Radoslaw Majewski.
Cohen then scored with a free-kick which crept in to give Forest a victory which their overall display deserved.
Manager Billy Davies stated after the draw at Watford at the weekend that his players were not good enough to be a top-six side, but after a disciplined performance they remain in the hunt.
Aidy Boothroyd's visitors had their moments but could not find a way back into the game in the closing stages.
Coventry striker Marlon King made just his second start since his release from prison against the club he cost £950,000 almost exactly seven years ago and was the main threat throughout.
At the other end Lewis McGugan sparked danger inside the first minute with a fabulous ball in from the left which Coventry goalkeeper Keiren Westwood had to be strong to get to in front of the lurking Robert Earnshaw.
The home side had made a sharp start and strike partner Nathan Tyson should have done better with a shot from the edge of the box moments later.
But Boothroyd's side arrived in high spirits themselves, despite a home defeat to Leeds on Saturday, and carved open a chance early on when King linked well with Richard Keogh, who crossed just in front of Lucas Jutkiewicz.
Forest had bossed a 20-minute spell but it was the visitors who took the lead in the 35th minute.
Jutkiewicz won a free-kick on the edge of the area and when Sammy Clingan - another former Forest player - whipped the ball into the heart of the area Chambers was unfortunate to see his header bounce down and into the top corner.
Aron Gunnarsson was challenging for the header and may have scored if Chambers had not intervened.
Forest did not deserve to be behind and were not for long as Majewski struck a fine goal in the 41st minute.
Ryan Bertrand laid the ball off to him 30 yards from goal and the Pole took one touch before arcing a dipping shot over Westwood's head and into the far corner.
Coventry came close to regaining the lead at the start of the second half from another Clingan dead ball.
His corner caused havoc in the area and Keogh looked certain to convert at the back post but his header somehow stayed out.
Tyson came close to a freak goal when he charged down a clearance from Westwood before the goalkeeper recovered to gather the ball, but Forest did strike in unusual circumstances in the 62nd minute.
Cohen won a free-kick wide on the right and bent in a dangerous ball to the back post which evaded Earnshaw's diving attempt and found the far corner.
King then had his first chance of the evening when he flashed a header wide from an excellent ball in from substitute Michael Doyle, and the striker then hit the side-netting at the end of an incisive breakaway.
King had a final effort saved by Lee Camp but Forest held on to see out the win which lifts them to within a point of Coventry and the play-offs.
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