Gary Mercer made five changes to the side nilled by Rotherham last week and he saw an immediate response from his team as they crossed for the opening try of the game after just 12 minutes.
The move was started by Christian Georgiou on half way before the ball was fed wide and David Doherty marked his return from injury with a try.
The try took the winger to 38 in his career for his home-town club and into clear second place in the all-time list of try scorers, one ahead of Dan Scarbrough and with just Tom Biggs ahead of him now.
Harry Leonard added the conversion from wide out to give Carnegie a 7-0 lead. However, it was Bedford who maintained pressure well enough to threaten next as they kept it tight in the forwards before scrum-half Darryl Veenedaal spotted a gap at the ruck and darted through to score an unconverted try.
Yorkshire had a chance to increase their lead when they were awarded a penalty 25 metres out but Leonard pushed his effort wide.
The visitors were then reduced to 14 men when second row Matt Smith was shown a yellow card. Carnegie defended well but they could not deny Bedford taking the lead as Will Sharp kicked a penalty from 30 metres out.
Bedford were awarded back- to-back penalties and, just as Smith returned from the bin, they camped themselves on the Yorkshire line.
Carnegie initially did well to repel the threat. However, when the ball was spread back into centre field, Bedford full-back Jim Wigglesworth cut back inside to score between the posts.
Sharp added the simple conversion but then missed a penalty from not much further out to keep the visitors in the game at 15-7 and with 25 minutes to go.
Carnegie battled hard but were not able to build any real pressure on the Bedford line and needed a late penalty from Leonard for a losing bonus point.
Bedford: Wigglesworth, Tapley, Dodge, Hakalo, Adamson, Sharp, Veenendaal, Parilli-Ocampo, Clare, Boulton, Howard, Tupai, Blanchet-Dufresne, Fenton-Wells, Vandermolen. Replacements: Saunders, Williams, Gulliver, Buggea, White, Mordt, Staff.
Yorkshire Carnegie: Holmes, Georgiou, Vickerman, Clarke, Doherty, Leonard, Dudman, Imiolek, J. Walker, Tideswell, Hannay, Smith, Williams, C. Walker, Burrows. Replacements: Nilsen, Harris, Hill, Edwards, Barnard, Pilgrim, Goss.