The hosts were the better side in the second half and deserved the win overall, although Jack Marriott should have earned a point for Derby when he scuffed a 76th-minute shot.
Derby started the brightest and could have taken the lead in the first 10 minutes when Bogle drove forwards from right-back and released Tom Lawrence, who was heavily booed with every touch.
Lawrence got his shot away but Brice Samba managed to get a strong hand to the ball to push it away to safety from 12 yards.
Derby suffered an early blow when central defender Matt Clarke limped off five minutes after going down with what appeared to be a knee injury.
Forest settled and tested Roos when Sammy Ameobi drove an effort at the Dutchman's near post.
Ameobi then had Forest's best effort of the first half when his stinging drive was pushed out by Roos in the final minute of the opening period.
Forest were much the brighter side immediately after the break and took the lead soon after Ameobi had drilled a yard over.
It came from Bogle's terrible error when his square ball was picked off by Grabban, who slotted past Roos first time from the edge of the box before the keeper could set himself.
The hosts might have gone further ahead when Ameobi saw two efforts blocked before Joe Lolley's left-footed effort was deflected wide.
Lawrence was perhaps lucky to stay on the pitch for his 70th-minute lunge on Oliveira Ribeiro, for which he was only booked.
Derby had rarely threatened in the second half but should have equalised when Marriott scuffed his effort from six yards with Samba able to dive on the ball.