The win moved the Reds above their Midlands rivals and out of the Championship relegation zone.
Following a forgettable first half, Forest were much-improved after half-time and McCleary gave them a deserved lead with a superb solo effort after 74 minutes.
Substitute Findley sealed the win with four minutes remaining to pile more misery on Coventry, who have now lost their last nine away matches in the league.
For a long time, a goalless stalemate looked inevitable. Bottom side Doncaster are the only team to have scored fewer home goals than Forest this season, while Coventry are the lowest scorers in the division away from home.
Chances were at a premium in the first half, which ended with Forest being booed off by their supporters.
Alex Nimely shot wide from 25 yards early on. The same player had the ball in the net in the 17th minute but it did not stand as Gary McSheffrey was penalised for handball.
At the other end, Marcus Tudgay sent a shot wide before the break.
Forest boss Steve Cotterill responded at half-time with a double substitution. Findley and fellow striker Dexter Blackstock were introduced with Tudgay and midfielder Scott Wootton making way.
The changes had the desired effect, with the second half more of a spectacle.
Findley was involved immediately, winning the corner from which McCleary shot wide.
Forest then hit a post in bizarre fashion. Chris Gunter's cross struck the unknowing Blackstock as he challenged the goalkeeper and the ball bobbled towards the net but came back off the upright.
A brave piece of goalkeeping from Joe Murphy then denied McCleary as he dived at the feet of the Forest forward.
Forest were firmly in the ascendancy but it was Coventry who almost took the lead when Gary Deegan unleashed a rocket of a shot that flew narrowly past the top corner, striking the stanchion behind the goal.
Cotterill's team were back on the attack straight away and only a superb save from Murphy prevented them from taking the lead.
McCleary's driven cross was flicked goalwards by Reid and the goalkeeper somehow managed to turn it away.
However, there was little Murphy could do to stop McCleary's effort shortly afterwards.
McCleary carried the ball from inside his own half, cut infield and then squeezed a left-footed shot inside the near post from 18 yards out.
That seemed to knock the wind out of Coventry, who were left totally deflated when Findley struck in the 86th minute.
The USA international controlled Andy Reid's cross on his chest then slotted his shot under Murphy to give Forest a first win at the City Ground since November 19.