Tries through David Doherty and Nathan Thomas in the opening 20 minutes proved crucial and laid the platform for the Headingley Carnegie outfit’s hard-earned success.
The scores helped send Leeds in at the break with a 17-6 advantage – and although Bristol initially rallied strongly after the restart, the Tykes ground out a win which director of rugby Phil Davies had demanded.
The first half of Leeds’ campaign was desperately sub-standard – and defeat here would have been close to disastrous.
But they are steadily improving, and this was their fifth victory from their last seven home games – suggesting the Tykes are capable of preserving their Premiership status.
Leeds made a vibrant start and were rewarded after four minutes when Roland De Marigny opened the scoring with a routine penalty.
The Italian, playing at fly-half with Gordon Ross away on Scotland duty, had the opportunity to double his side’s advantage shortly afterwards but sent a 35-metre penalty wide of the posts.
However, Leeds remained dominant – and a bewildering array of inter-passing saw them to claim an exhilarating try after 14 minutes.
Justin Marshall was the orchestrator, the 81-times capped New Zealand scrum-half finding Tom Palmer to his right in midfield with a swift offload.
The Tykes second row kept the move flowing with another neat pass to Chris Bell, who in turn found Doherty in space on the right flank.
The 19-year-old displayed outstanding speed to outstrip the visiting defence down the right flank and cross the line for his first ever Guinness Premiership touchdown.
De Marigny added the extras, and there was precious little respite for the visitors as Leeds continued to attack with purpose.
Midway through the opening period, Bristol had second row Nathan Budgett sin-binned for a professional foul in the ruck, and seconds later Nathan Thomas forced his way over the whitewash from close range to extend the Tykes’ lead.
It was a first try for Thomas in Leeds colours, but two penalties in as many minutes from Jason Strange shortly after the midway point in the half hauled Bristol back into contention at 17-6.
The momentum remained with Bristol immediately after the restart when Lee Robinson collected a pass from opposite wing David Lemi and darted through a gap in the Tykes’ rearguard to cross the line.
Strange again converted to reduce the arrears to 17-13, but the Bristol fly-half was left disappointed moments later when he missed a penalty which would have edged his side to within one point of the hosts.
The impressive De Marigny punished Strange’s profligacy soon afterwards with two well-executed penalties to take Leeds 26-13 ahead.
Tommy Hayes’ 66th-minute penalty was nothing more than consolation for Bristol, as De Marigny soon completed a personal 16-point haul.