Richard Keogh had a good early chance for the visitors but his header from 10 yards was mistimed and didn't trouble Robert Olejnik.
Derby continued to look dangerous and a long-range strike from Theo Robinso was only a couple of inches away from making the breakthrough.
Peterborough should have taken the lead against the run of play when Saido Berahino found space inside the box but his effort was wide of the far post.
The home side did manage to open the scoring through Bostwick, with the midfielder volleying home from 20 yards.
Adam Legzdins ensured that Derby didn't fall further behind before the break by superbly tipping over Berahino's curled effort.