Hibs had a goal back immediately when Elding appeared to divert the ball in with either his chest or his shoulder from Felix Passlack's cross. A VAR check for hand-ball followed but the goal was allowed to stand.
The next intervention from the video assistant referee went against Hibs, who had been awarded a penalty when Klidje went down under Lewis Neilson's challenge.
The decision was changed to a free-kick to Hibs just outside the area, an award which was blasted wide by Martin Boyle, who also wasted a glorious chance when his close range shot was too close to Scott Bain.
Falkirk were happy to sit on their lead in the second half as a fairly toothless Hibs failed to seriously them, with the exception of a fine shot by substitiute Jamie McGrath which Bain brilliantly tipped round a post.
It means Falkirk move up to ninth with their first win of any kind in Leith since 2014.
Life looks altogether bleaker for Hibs, however, with this damaging defeat leaving them bottom and without a point from Reedijk's first two games in charge.