Jersey were denied the upset of the Championship season so far as leaders Bristol grabbed two late tries to come from behind and extend their winning run.
The islanders led 17-15 after tries from forwards Sam Lockwood, Joe Buckle and Ryan Hodson.
But Nicky Robinson's last-gasp try ensured his side took full points.
Marco Mama earlier crossed the whitewash twice for the visitors to add to a penalty try.
Jersey had gone into the game hoping to avoid an eighth consecutive defeat, while Bristol had won all 16 of their matches in cup and league.
|Jersey's winless run and Bristol's winning streak|
|Jersey are without a win in any competition since 8 November, when they beat Plymouth Albion, while Bristol have won all 17 of their competitive fixtures this season.|
Matt Morgan's penalty put Bristol into an early lead in strong winds at St Peter, but prop Lockwood crashed over from a line-out drive to put the islanders in front.
The West Country side were back ahead, though when referee Darren Gamage awarded a penalty try as the maul looked poised to cross the line.
At the other end Bristol were reduced to 14 men with Ben Mosses sent to the sin bin, and Jersey made the advantage count as hooker Buckle found space down the right to score and make it 10-10 at the break.
The driving maul from the line-out seemed the flavour of the day for both sides, and Bristol used it to good affect as they went back in front as Zimbabwean flanker Mama scored his first.
Jersey still had hope of claiming one of the greatest results in their history though, and when prop Gaston Cortes was shown yellow the league leaders were down to 14 men again.
The hosts capitalised and drove over from another line-out, with flanker Hodson credited with the converted try which put Jersey in front at 17-15 with 10 minutes remaining.
Bristol though have got used to winning and showed real character in the tense closing stages.
Mama's second score of the afternoon gave them a narrow lead before Robinson spotted a gap in the Jersey defence with the final move to give his team the bonus point and, in the process, snatch the losing bonus point away from the Channel Island team.
Bristol remain two points clear of title-rivals Worcester at the halfway point in the Championship season, while Jersey drop to 10th spot.
Jersey: Pointer; Foster, Bishop, Robling, Locke; Penberthy, Dudley; Lockwood, Buckle, Williams; Phillips, Campbell; Rae (capt), Hodson, Haining.
Replacements: Selway, Lancuba, Herriott, Markham, Noone, Glynn, Bentley.
Bristol: Slowik; Short, Tovey, Mosses, Lemi (capt); Morgan Hampson; Traynor, Brooker, Perenise; Evans, Townson; Mama, Robinson, Eadie.
Replacements: Lawrence, Cortes, Glynn, Koster, Baldwin, Robinson, Amesbury.