The win sees Bristol move two places up to eighth on the table, while Leeds still bottom.
Wing David Lemi and second row Roy Winters scored a try apiece, while fly-half Jason Strange kicked 22 points as Bristol moved two places up to eighth on the table, while Leeds remained stuck at the bottom.
Bristol applied the pressure from the start and Strange kicked his first penalty in the fourth minute, Leeds tried to hit back and a typical Andre Snyman bash through the middle was well contained by the Bristol defence, and several minutes later Bristol extended their lead with another Strange penalty.
Leeds narrowed the gap when fly-half Gordon Ross converted a massive 45-metre penalty, but Strange slotted his third a few minutes later.
Ross missed a second attempt from just inside the Bristol 10-metre line, and he was made to rue the miss when Lemi again showed pace and determination which eventually led to more points from Strange.
Leeds tried to stretch the Bristol defence as wing Dave Doherty broke down the left, but he was stopped 10 metres from the line by centre Sam Cox and full-back Vaughan Going.
Ross landed his second penalty, but Lemi scored his try at the very end of the first half. Centre Sam Cox barged up the middle and after some quick ball, scrum-half Shaun Perry lofted a long pass to Lemi who dived over in the corner. Strange converted, giving Bristol a 19-6 lead at the break.
Bristol started the second much as they had started the first with Strange kicking his fifth penalty after six minutes.
Ross missed his third penalty and then had salt rubbed into their wounds when skipper Justin Marshall was yellow carded as Strange notched up penalty number six.
Bristol were not finished yet and with Lemi held up over the line, it was the supporting Winters who crashed over to end their day in style.