A header and a follow-up shot after a save from the ex-Millwall striker looked to have won it for the Robins after they came from behind to lead.
But in stoppage time Clarke Carlisle scrambled an equaliser for Burnley after an Ade Akinbiyi shot had been partly blocked. It was a dramatic finish to a match which was for so long had been poor entertainment.
City's manager Gary Johnson was left to congratulate himself on two match-changing substitutions, sending on Byfield at half-time and then throwing on Hungarian centre-back Tamas Vasko as a striker to harass Burnley.
Both switches worked perfectly to rescue a disjointed game which took so long to settle.
In the first half Burnley's raids were failing in the final third while the only real threat to their goal came when City's left-back Jamie McAllister swung over a long cross.
Marvin Elliott, also a signing from Millwall, outjumped the Clarets defenders but glanced his header wide.
It was Burnley's Wade Elliott who promised most and one run through the middle drew a memorable tackle from Louis Carey.
The timing of the City captain was not always as immaculate and three fouls in five minutes later carried him uncomfortably close to a booking.
There was a card for Burnley's Michael Duff for bringing down Mike McIndoe who then produced a smothering save from former Palace keeper Gabor Kiraly.
Searching for shape and pace both managers made a change at half-time and Burnley boss Steve Cotterill was the first to gain a reward.
Six minutes into the half Elliott, coming in from the left, hit his shot against the foot of the post for Andy Gray to run in the rebound for his sixth goal of the season.
That was the end of the scoring until the final ten minutes when almost all of the game's drama occurred.