Kings dominated the first half, controlling the first quarter of an hour with a Mitchell Weiss shot deflected and a Dean Hitchcock fizzer that Maxime Fondup Siwe could not hold. Weiss netted the rebound but was ruled offside.
At the other end, Xavi Comas had to resort to some punched clearances, but the home side took the lead with a glancing Connor Toomey header from a Dean Hitchcock corner on 18 minutes.
Weiss then pounced on a poor Josh Beech clearance to drill the ball home and Kings were now 2-0 up.
Meanwhile Christian Lester unleashed a shot just past the post for Bedford’s best chance of the opening period,
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In the first minute of the second half the Eagles grabbed a lifeline when Toomey made a hash of clearing his lines and Josh Bamford gratefully lashed the ball past Comas.
This breathed life back into the Eagles and they enjoyed their best 10 minute passage of the game.
But Weiss restored the buffer when he latched onto a long ball, was forced wide on the byline by the attentions of two defenders but he beat the pair and the keeper for a goal of sheer persistence.
Bedford had their moments, with James Peters missing a golden chance to reduce the deficit again, but as the temperature continued to drop below freezing point and the large crowd became concerned about the implications of frostbite, Kings saw out a comfortable victory to keep them on the edges of the play-off zone.
Bedford are due to visit St Ives in the league on Tuesday.