They saw off the challenge of Ulster A 22-17 in tricky conditions with their four tries securing the extra bonus point.
The hosts pinned the Blues back during the early exchanges but Bedford soaked up the pressure and cleared the early danger.
And it was the visitors who scored first, Phil Boulton touching down from a driving maul forced after Jake Sharp sent a penalty into touch deep in Ulster’s 22.
However on a sticky pitch at Deramore Ulster fought back. From a lineout Jonny McPhillips passed wide to winger Jacob Stockdale who barged over, with McPhillips adding the extras to make it 7-5
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Jordan Burns came close for Bedford when he spotted a gap with a quick penalty just inside his own half, but he was hauled down just short.
However they were ahead soon after when Harry Wells broke down the wing and he passed to Dean Adamson outside him to score, Sharp converting.
McPhillips scored a 39th minute penalty for Ulster to reduce the deficit to 12-10 at the break.
Bedford extended their lead to 17-10 early in the second period. Nick Fenton-Well exploited the space before passing to Tom Griffiths who fed Henry Staff to cross the line for an uncoverted try.
But if they thought the game was theirs for the taking Ulster had other ideas. On 56 minutes Rodger McBurney fed Sam Windsor to cross the line and McPhillips added the extras to make it 17-17 going into the final quarter.
After fending off waves of attacks fromm the home side Bedford pushed forward themselves and it was Steffan Jones who went through to grab the fourth try.
Bedford’s defence had to stay alert as Ulster’s powerful driving maul bore down on the tryline, the Blues holding firm and disrupting the attack to ensure they returned to Bedfordshire with the win.
They are back in action on Saturday, January 23 at home to Scarlets Select in the final Pool 2 game of the British and Irish Cup.
Bedford: S Jones, S Blanchet, H Staff, T Griffiths, D Adamson, J Sharp, J Burns; T Williams, C Clare, P Boulton, H Wells, M Flanagan, J Blanchet, N Fenton-Wells (capt), J Conlon. Replacements: S James, C Parilli-Ocampo, P Tupai, B Alo, P White, M Dorrian, O Dodge