A brave second-half fightback was not enough to earn Bristol a second Championship play-off win.
A brace of tries for Bedford's Brendan Burke in five first-half minutes gave the home side a 17-3 lead at the break.
Two Adrian Jarvis penalties and Redford Pennycook's converted try narrowed the gap to just one point with 25 minutes still left at Goldington Road.
But a last-minute drop goal attempt from Jarvis went agonisingly wide and Bedford hung on to go top of Pool A.
Bedford have now won their opening two play-off games by just a point following last Saturday's 10-9 win at Plymouth Albion.
They now travel to Cornish Pirates on Easter Sunday (4 April, 1500 BST).
Bristol, who beat the Pirates 19-12 in their first play-off match, are in action on Good Friday against Plymouth (2 April, 1500 BST).
Bedford chairman Geoff Irvine told BBC Three Counties Radio:
"I can't remember being that emotional since drawing at Otley.
"What a wonderful display from the lads. It was a bunch of guys that put their bodies on the line.
"I'd like to think that over the last 10 years I have seen everything, but I have not seen a more magnificent display than we saw from the 22 players wearing Bedford rugby shirts today."
Bristol head coach Paul Hull told BBC Radio Bristol:
"I'm very disappointed but when you go in at half-time 17-3 down in these kind of fixtures you can't expect to claw your way back."
"It was just lapses in concentration but it was a hammer blow because they scored twice in close succession."
"We need to win next week. There are no grey areas and we need to push on and try and finish this group on top."
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