Watched from the stands by new signing Paul Ince, who agreed a one-year deal with the Robins this week, they almost had the perfect start inside five minutes when Ricky Shakes made good progress down the right flank to cross for Aaron Brown whose far post header was smartly saved by keeper John Danby.
The visitors certainly looked lively going forward and thought they had taken the lead on 22 minutes when Brown pounced on a rebound to lash the ball home, only for the official to rule the effort out for a handball.
Charlie Comyn-Platt had an opportunity on 33 minutes with a difficult volley which he couldn't quite keep down below the crossbar.
Chester saved their only real threat of the first half deep into added time when Jon Walters headed on to Drewe Broughton, but the forward's lofted shot cleared both the keeper and the bar.
Having lived comfortably with their high-flying opponents in the first half, Chester were hit three minutes after the interval when the Robins took the lead. A free-kick from the left by Jack Smith managed to find Lee Peacock at the far post and he made no mistake firing the ball past Danby at his near post.
Chester tried to hit back and went close with Laurence Wilson striking a fine left-footed effort on 65 minutes, which appeared destined for the bottom corner of the net, until keeper Petr Brezovan produced a superb save at full stretch to somehow tip the ball aside.
The home side were given hope on 70 minutes when their opponents were reduced to ten men, with Jerel Ifil being dismissed with his second yellow card after foolishly kicking the ball away when a free-kick had been awarded.
But that hope was extinguished just five minutes later when Peacock added a second, nodding the ball home from Andy Nicholas' free-kick to secure the points and send the 800 travelling faithful into raptures.