In an end-to-end match the Pirates could do little more as Williams pulled off two stunning saves before a further two chances were cleared off the line.
Within the space of two weeks the Cumbrians have gone from play-off hopefuls to staring at a relegation dogfight and sub Scott Dobie summed up the transition with a slapstick miss from a yard out.
The hosts were quick out of the blocks with a flurry of early corners before Kavanagh composed himself 30 yards out and rifled in an unstoppable dipping drive in the fourth minute.
But as so often this season Carlisle failed to hold on to their lead and within ten minutes Bristol were back on terms.
A hopeful punt upfield from Ryan Green found Jeff Hughes in the area and the Northern Irishman turned easily inside the flat-footed Michael Liddle and fired past Williams.
Carlisle were dealt a blow ten minutes before the break when top-scorer Danny Graham was forced off injured, before Hughes almost added to their misery when he ran through for a certain second, only to be denied by a fine Williams block.
The Carlisle keeper pulled off another top save to deny Darryl Duffy after he drifted past two defenders to hit a dipping volley that Williams somehow tipped over at full stretch.
Rovers' danger man Rickie Lambert then saw his weak bouncing header cleared off the line by Peter Murphy and five minutes later the scenario was repeated when Tony Kane hacked clear from the striker's miscue.
Within 60 seconds the game should have turned in Carlisle's favour, but when Liddle's wicked cross reached the far post, Dobie somehow failed to connect when almost on the goalline, leaving both sides with a point.