It took the visitors only three minutes to break through the Forest defence. Coventry's Steve Warnock crossed into the box and the unmarked Cameroon international Patrick Suffo smashed a low shot past Darren Ward to give the visitors an early lead.
Midway through the first half Forest nearly got the equaliser they had been looking for.
A 20-yard shot from Michael Chopra, who is on a month's loan from Newcastle United, beat the keeper but fell inches wide.
With five minutes left in the first half Forest's Marlon King stretched for a deep cross and although he did make contact he could not find the target.
Forest went into the half-time interval one goal down and are now staring at the reality of Second Division football next season.
Hart made a change at the beginning of the second half with Gareth Taylor coming on for Danny Sonner.
This was a tactical change with Hart playing a 4-3-3 formation. This made an immediate impact and with only two minutes gone in the second half, substitute Taylor broke through and had a powerful shot saved by a vigilant Scott Shearer.
An improving Forest turned on the pressure and again threatened to equalise on the hour mark as Taylor got a head to Andy Reid's powerful cross, but failed to guide the ball with any accuracy.
Although Coventry didn't threaten Forest's goal after the third minute, some credit has to be given to their back four who stood up to everything a desperate Forest had to throw at them.