Facing their biggest test of the season, Neathwere struggling to even raise a side for the match due to a reported 14 players on the injury list.
As an emergency procedure they had to draft in Wales Rugby League international winger Lewis Reece from South Wales Scorpions, along with four other new players on the bench.
The Blues took the lead in the second minute after James Pritchard kicked a 40 metre penalty after Neath were penalised for offside.
Aaron Bramwell had the chance to level things four minutes later but his long range penalty kick went wide.
Those were the only penalty attempts of the game. After that, free flowing open rugby was the order of the day, mainly from the exciting away side.
Bedford increased their lead on 12 minutes thanks to superb move that ripped the Neath defence apart. Ben Gulliver laid the ball off nicely to Scott Spurling who in turn fed Josh Bassett who raced in. Pritchard converted the score.
Reece almost marked his Neath debut with a try but Chris Morgans' pass to him on the wing was forward.
But it was Reece's former Scorpions team mate, Kevin James, who finally broke Neath's duck on 24 minutes and it was a score the home side deserved for their constant pressure. From a scrum, Matthew Rees and Morgans set up the opportunity and James used his speed and strength to power through. Bramwell converted.
Bedford addressed the balance straight away when, after winning a penalty, Jake Sharp chipped through for James Stephenson to receive and ground in the corner.
Pritchard missed the conversion but made up for it by adding a third Blues try in the same right hand corner, two minutes before the break following an excellent passing move before adding the conversion himself.
The Blues weren't finished yet in the first period and Bassett broke through the for a fine individual try which Pritchard duly converted to give Bedford a 29-7 lead and a valuable bonus point.
They then took just three minutes to get on the board in the second half. Stephenson broke through the back-line and grounded under the sticks. Spurling then increased the Blues lead with a try from a rolling maul five minutes later. Pritchard converted both scores.
Neath hit back on 55 minutes when Morgans set up James to dive home for his second try of the match.
But Bedford finished on a high with three tries in the last 15 minutes.
Bassett romped through for his third and Bedford's seventh on 65 minutes before Mark Kohler scored in the corner.
And a final Pritchard try in the corner completed the rout.
Neath: 15 Aaron Bramwell; 14 Kevin James, 13 Ryan Evans, 12 John Leyshon, 11 Lewis Reece, 10 Kyle Davies 9 Chris Morgans; 1 Nicky Downs (captain), 2 Rhodri Clancy, 3 Marc Jones; 4 Rhys Jones, 5 Jonny Griffiths, 6 Leon Ward, 7 Matthew Rees, 8 Steffan Jones.
Replacements - 16 Tom Price, 17 Alun Wyn Davies, 18 Owain Ford, 19 Tom Davies, 20 Simon Pengelly, 21 Gareth Davies, 22 Lloyd Phillips
Tries: James (24, 55)
Cons: Bramwell (24)
Bedford: 15 James Pritchard (captain); 14 James Stephenson, 13 Mark Atkinson, 12 Ian Vass, 11 Josh Bassett; 10 Jake Sharp, 9 Darryl Veenendaal; 1 Darren Fearn, 2 Scott Spurling, 3 Dan Seal; 4 Mike Howard, 5 Ben Gulliver; 6 Alex Rae, 7 Darren Fox, 8 Joe Vandermolen.
Replacements - 16 Chris Locke, 17 Ben Cooper, 18 Greg Gillanders, 19 Tom Armes, 20 Ali Price, 21 Brendan Burke, 22 Mark Kohler
Tries: Bassett (12, 40, 65), Stephenson (27, 43), Pritchard (38, 79), Spurling (48), Kohler (69)
Cons: Pritchard (12, 38, 40, 43, 48, 65, 69)
Pens: Pritchard (3)
Referee: Cammy Rudkin (Scotland)
Attendance: 700 approx