Dykes blasted home from the penalty spot nine minutes into the second half at the Kiyan Prince Foundation Stadium after being brought down by Tobias Figueiredo.
The breakthrough came for the hosts when Lee Wallace lifted the ball forward and Dykes, who this week scored his first international goal, got behind to Figueiredo and was then tripped by the defender.
It was exactly the start to the new campaign Forest did not want after missing out on the play-offs in extraordinary circumstances at the end of last term, when a 4-1 defeat at home against Stoke saw them drop out of the top six.
Rangers' goal sparked a reaction from Forest and Ryan Yates hit the bar before Grabban was guilty of a glaring miss.
Grabban did well to thread a pass through to Yates, who turned nicely and slammed a shot against the woodwork from an awkward angle.
Grabban should have brought Forest level a few minutes afterwards but miscued from close range after being teed up by substitute Sammy Ameobi.
And Grabban missed another opportunity seven minutes from time when he hooked a left-footed shot wide of the target – again after being set up by Ameobi.
QPR have had a difficult summer themselves, with star man Ebere Eze having left for Crystal Palace and uncertainty surrounding the future of winger Bright Osayi-Samuel, who is in the final year of his contract and has rejected offers of a new deal.
Osayi-Samuel was selected and caused Forest problems, especially in the first half, although QPR struggled to create clear-cut chances and were on the back foot for large spells after the goal.
But they defended resolutely and Dykes, a recent signing from Livingston, can feel pleased with his start to life in the Championship.
He almost netted a second when he flicked the ball over the bar after being found by Chair, whose stoppage-time goal sealed Forest's fate.