The bonus-point away victory also kept Carnegie in the hunt for a place in the knockout stages of the British & Irish Cup.
Redpath said: “There might have been some lingering doubt that it had been a big week for me and the players if we hadn’t have performed so the boys put that to bed.
“It was really hard for me to tell them. I thought it would have been a lot easier, but we’ve got to look forward and see what comes out in the next 10 or 12 games.”
After an utterly forgettable first 20 minutes, Carnegie opened the scoring when prop Charlie Beech burrowed his way over from close range. The recalled Warren Seals added the extras from 15 metres out.
Bedford were reduced to 14 men when hooker Dan George was sent to the sin-bin for a shoulder in the back of Phil Nilsen.
Carnegie immediately made their man advantage count from the resulting penalty.
The visitors caught and drove from five metres out and Rob O’Donnell dotted down, a try which Seals duly converted.
Carnegie were perhaps a little fortunate not to have been reduced to 14 men when Dan Sanderson appeared to raise a hand to Justin Blanchet just before half-time, but referee Hamish Smales just gave the lock a talking to.
Another catch and drive brought about Carnegie’s third try. This time, Ollie Stedman was the beneficiary of the visitors’ dominance up front.
Seals maintained his 100 per cent record for the afternoon with his third conversion and the South African fly-half’s fine afternoon continued when he finished off a free-flowing Carnegie move in the corner, though he could not add the extra two points.
Possibly having thought the game was won, a lapse in the visitors’ defence allowed Ollie Dodge to weave his way through a couple of tackles to get the Blues on the scoreboard.
Jake Sharp’s conversion cut the deficit to 19 points.
Yorkshire thought they might have had a fifth try, and a fourth through a driving maul, but this time referee Smales adjudged Bedford had done enough to hold the ball up.
The Blues scored their second try when replacement Tom Whiteley took a quick penalty tap that caught Carnegie napping, and, despite slipping over just inches in front of the line, he was still able to recover his composure to score.
Sharp missed the conversion meaning the gap remained at 14 points.
Carnegie were reduced to 14 men when Richard Mayhew was sent to the sin-bin for leaving out a trailing arm as a Bedford back was trying to chase a chip through.
But Carnegie repelled all that Bedford could muster with a man advantage and Seals completed the Yorkshire side’s win with a penalty.
Bedford Blues: O’Conor, Tapley (Robinson 36-40), Dodge, Griffiths, Clements-Hill, Sharp (c) (Dorrian 67), James (Whiteley 51), Parilli-Ocampo (Culverhouse 51), George (Robinson 74), Walker (Judgeman 50), Carrick-Smith, Hodge (Buggea 55), Blanchet, Adams, Hill.
Yorkshire Carnegie: McColl, Goss, Gray, Wright (Casson 50), Prell, Seals, Green, Beech (O’Donnell 78), Nilsen (M. Mayhew 70), O’Donnell (Cusack 55), Sanderson (Smith 51), West Mayhew (c) (Staples 79), Saull (Bainbridge 62), Stedman. Unused: Stegmann.
Referee: H Smales (RFU).