The second half was better for England and Jonny May's introduction off the bench followed by that of Tom Homer seemed to boost the attacking options.
A spell of pressure finally paid off as flanker Jamie Gibson took a short pass from Burns to fight his way to the line for the game's only try.
Burns missed with the conversion and the usually reliable Homer kept it interesting as he promptly missed with two penalty attempts including one from bang in front.
A drop goal from Rory Clegg eased the nerves as time ticked on with England hanging on to a narrow 11-6 lead but this was promptly cancelled out by a French penalty.
However, with the game in injury time, Burns slotted a penalty to close out the game and take England into the semi finals.