The Reds, who hadn't lost away from home, have now suffered defeats in their last two outings on their travels as Coventry ended a run of three straight draws.
Forest manager Billy Davies shuffled his pack after the weekend win over Sheffield Wednesday with Robert Earnshaw dropping to the bench.
And the visitors had the first chance of the opening period as Chris Cohen played a quick free-kick to Joe Garner, but his shot was blocked by Coventry keeper Keiren Westwood.
Forest looked good in possession but Chris Coleman's team were limiting their goalscoring chances and the Welshman's side almost took the lead after 16 minutes but Aron Gunnarsson headed over Sammy Clingan's cross.
But Coventry did open the scoring eight minutes later. Clinton Morrison held the ball up superbly on the left hand side before crossing low for Richard Wood.
The big defender's shot was stopped right on the line by Lee Camp and Eastwood was on hand to blast into the roof of the net for his seventh of the season.
Forest did begin to put a little more pressure on the home team's goal as Paul Anderson volleyed wide before Garner headed off target with Westwood off his line.
The Sky Blues came close to doubling their advantage after 37 minutes but Michael McIndoe's shot was well saved by Camp.
Forest upped the tempo in the second half and had some good opportunities. Guy Moussi headed wide from a cross from Radoslaw Majewski when he should have scored.
Then the Polish midfielder then had two long-range shots of his own but both were off target.
Having weathered a bit of a storm, Eastwood had a great chance to double City's lead after great forward play from Morrison but shot over the bar.
In the 71st minute, referee Tony Bates was forced off with an injury and female official Amy Fearn took over the duties.
The home side almost sealed it in the 87th minute after a fine move involving Eastwood and Carl Baker set up Gary Deegan but he shot into the side-netting.