Bristol face an all-or-nothing showdown with West country rivals Gloucester at Ashton Gate on Saturday for a Guinness Premiership play-off spot, after they beat a weak Leicester Tigers team 30-13 on Tuesday.
While trophy treble-chasers Leicester remain second behind Gloucester, Bristol climbed two places into third.
Victory over Gloucester on Saturday will confirm a play-off place, while Wasps can only realistically keep alive their European and domestic double dream by defeating Welford Road hosts Leicester in a dress rehearsal for next month's Heineken Cup final.
Bristol though, will not be concerned with events in the Midlands, as they continue to defy the Premiership pundits, many of whom tipped them for relegation just eight months ago.
Their 14th win of the league campaign - a figure only Gloucester can match - was underpinned by an impressive display from Bristol's rookie number 10 Danny Gray.
Gray, who has played a support role to fellow fly-halves Jason Strange and David Hill this season, took his opportunity superbly.
He produced a high-class goalkicking performance in rainswept conditions, booting 20 points through six penalties and a conversion of lock Roy Winters' 12th-minute try.
Leicester, their squad resources stretched by a demanding late-season fixture list, competed robustly throughout, yet Bristol secured an emphatic success when substitute hooker Scott Linklater powered over in the closing minutes.
Tigers made 13 changes from the side that defeated Heineken Cup semi-final opponents Llanelli Scarlets last Saturday, with only wing Tom Varndell and prop Alex Moreno remaining in a team captained by lock Leo Cullen.
Bristol, also firmly in the play-off picture, were without injured England scrum-half Shaun Perry (calf muscle) and wing David Lemi (hamstring), who both failed late fitness tests, which meant respective starts for Greg Nicholls and Brian Lima.
But there was no obvious sign of disruption to Bristol as they posted 16 points during the opening quarter, recovering impressively from conceding a soft try.
Gray booted penalties either side of Tigers' opening try, which arrived after Matt Cornwell charged down Gray's attempted defensive clearance, allowing his midfield partner Smith an easy run-in.
Humphreys converted, but Bristol rapidly regained the initiative when slick passing between flanker Joe El Abd and full-back Luke Arscott sent Winters galloping over.
Gray converted, then completed his penalty hat-trick after the Leicester front-row conceded a second three-pointer as they came under growing pressure from Bristol's scrummaging unit of Dave Hilton, Mark Regan and Darren Crompton.
Leicester had their moments during a lively first period, with Smith a considerable threat - even after being on the receiving end of a crunching Lima tackle - but Humphreys' goal-kicking proved a problem.
He missed two penalty chances, the second after Bristol number eight Andrew Blowers was sin-binned by referee Wayne Barnes for killing possession.
But Humphreys finally found his range on the stroke of half-time, cutting Leicester's deficit to 16-10 with Bristol temporarily a player short.
Gray, who will leave Bristol this summer and is likely to join National League One club Exeter Chiefs tomorrow, landed his fifth successful kick for a 19-10 advantage which Humphreys immediately cancelled out.
Bristol's forwards relished the close-quarter combat in a pursuit of territorial control, yet they found Leicester a tough nut to crack, and the home side saw El Abd sin-binned for pulling down a maul as the game entered a critical phase.
Tigers, despite trailing for most of the match, sensed an opportunity with El Abd off, and it took a 50-metre clearance by Arscott to relieve escalating pressure.
Arscott's kick gave Bristol a renewed lease of life, and with head coach Richard Hill using his replacements' bench effectively, the pack stepped up a gear, creating pressure for Gray to land two more penalties during a decisive four-minute spell.
Leicester could now only go in pursuit of a losing bonus point, but Bristol deservedly had the final say when Linklater sprinted over from 15 metres.
Tries: Winters, Linklater
Pens: Gray 6
Pen: Humphreys 2
Yellow cards: Blowers (34, Bristol, killing the ball), El Abd (59, Bristol, collapsing a maul)
Bristol: 15 Luke Arscott, 14 Lee Robinson, 13 Neil Brew, 12 Sam Cox, 11 David Lemi, 10 Danny Gray, 9 Shaun Perry, 8 Andrew Blowers, 7 Joe El Abd, 6 Alfie To'oala, 5 Gareth Llewellyn (c), 4 Roy Winters, 3 Darren Crompton, 2 Mark Regan, 1 Dave Hilton,
Replacements: 16 Scott Linklater, 17 Alex Clarke, 18 Sean Hohneck, 19 Nathan Budgett, 20 Greg Nicholls, 21 David Hill, 22 Brian Lima
Leceister:15 Johne Murphy, 14 Leon Lloyd, 13 Ollie Smith, 12 Matt Cornwell, 11 Tom Varndell, 10 Ian Humphreys, 9 Frank Murphy, 8 Jordan Crane, 7 Luke Abraham, 6 Tom Croft, 5 James Hamilton, 4 Leo Cullen (c), 3 Michael Holford, 2 James Buckland, 1 Alex Moreno
Replacements: 16 Gavin Hickie, 17 David Young, 18 Louis Deacon, 19 Brett Deacon, 20 Ben Youngs, 21 Tom Youngs, 22 Paul Burke