Instead, they completely lost the plot and were dragged deep into the relegation mire as West Hartlepool launched a magnificent revival to record their second League win in two seasons.
Against opponents who ran out of flankers with the late withdrawal of Jon Irons and Russell Earnshaw, Bristol's forwards established such early command that a huge victory in this basement battle looked a formality.
Liam Botham, making his League debut for West, had to wait 31 minutes for a touch. But he played his part in some spirited defence and Bristol's fumbling attempts to play expansive rugby eventually backfired.
Even without the England lock Simon Shaw, resting a back injury, Bristol had sufficient forward superiority to take telling advantage of the fact that West had to draft a scrum-half, Mark Roderick, and a hooker, Martin Challinor, into their back row.
Once several chances had gone begging, however, Bristol lost their way in the face of a West comeback inspired by the New Zealanders Ivan Morgan and Jamie Connolly.
West were much sharper behind the scrum and Connolly shrugged off feeble tackling to score in the 34th minute. John Stabler converted and then added a penalty to give West an unexpected 13-8 halftime lead.
For all the efforts of Jones, returning after an operation for a prolapsed disc, Bristol could not regain the initiative against opponents who found enormous strength in adversity.
West clinched victory through two more penalties by Stabler and are now only two points behind Bristol, whose desperation for victory saw them become increasingly frantic. They were pressing relentlessly at the close, but their lack of imagination and West's heroic defence kept them out.
The Bristol fly-half, Mark Tainton, had replied to Matt Silva's early penalty to raise his hopes of recording the 14 points he needed to break Bristol's individual points-scoring record of 2,034, held by Alan Pearn.
Tainton, however, missed a simple penalty during Bristol's period of dominance and West's resurgence ensured that his only opportunity came when Jones scored in the right corner.
West Hartlepool: M Silva; M Wood, J Connolly, L Botham, S John, J Stabler, P Harvey, W de Jonge, S Whitehead, P Whitelock, C Murphy, K Moseley (capt), M Roderick, M Challinor, I Morgan.
Bristol: P Hull; D Tiueti, K Maggs, M Denney, B Breeze, M Tainton, R Jones, A Sharp, M Regan, D Hinkins, P Adams, C Eagle, C Barrow, M Corry (capt), S Filali (N Temperly, 55).
Referee: C Rees (RFU).Reuse content