Ryan Shortland and Alex Tait both crossed for Newcastle while fly-half Jimmy Gopperth, playing his last game for the club before joining Leinster, kicked seven penalties.
Bedford stand-off Jake Sharp scored all of the Blues' points in an evenly-matched game, but it was not enough as Newcastle, who had a nine-point cushion thanks to their 18-9 win in the first leg, closed out promotion to replace London Welsh in the top flight.
After dominating the Championship all season and losing just one regular league fixture all term against Bristol, Newcastle were expected to ease their way back into the top flight. However, Bedford, who would not have been able to be promoted as their Goldington Road ground does not meet the standards required for Premiership stadiums, put up a determined fight in a match which was by no means a simple procession to promotion for Newcastle.
Bedford looked keen to unleash their backline early on, with full-back Ben Ransom trying to test the Falcons defence with a neat grubber kick intended for wing Josh Bassett, only for the ball to find touch. Bedford conceded a penalty from the subsequent lineout allowing Newcastle to clear, but their intention to stay on the front foot won them a second shot at goal in the 15th minute, which Sharp duly took.
With the gap closed to six points, the Blues fly-half needed no encouragement to go for the sticks midway through the first-half after Newcastle were penalised again for offside, with their early indiscipline almost wiping out their first-leg advantage.
One thing that was working for the Falcons, however, was their scrum, with Gopperth slotting a penalty straight through the middle of the posts following an infringement by Bedford. The dominance up front appeared to give the home side a confidence boost after their initial struggles, and wing Shortland settled nerves around Kingston Park when he crossed for a try after Newcastle's first real attacking play.
The Falcons worked through 13 phases and put the Blues under real pressure before Shortland crossed in the left corner, Gopperth missing the resulting conversion chance. But Bedford hit back immediately thanks to Sharp's quick thinking and a well-taken drop goal, before he traded two penalties with Gopperth to leave the visitors 18-17 ahead at half-time but still eight behind on aggregate.
Sharp cut the overall margin further with his sixth successful penalty early in the second half and Newcastle hit back immediately through full-back Tait. Tait crossed in the corner to edge the home side ahead before Gopperth's conversion attempt hit the bar. The fly-half was back on target from the penalty tee midway through the second half as Newcastle led 25-21, 13 points overall.
The Blues back line, which had looked so dangerous in the first period, lacked some of their earlier potency but Newcastle were lucky to avoid conceding a try when the ball appeared to strike Noah Cato's hand during an attack. Bedford were however, awarded a penalty for an earlier infringement, Sharp slotting the ball from in front of goal. Gopperth scored another penalty to keep the hosts' 13-point margin and added his seventh of the match with three minutes remaining to ensure Newcastle were home and dry.