Morecambe's draw at Coventry was enough to save them, but condemn Barnet to the drop for the first time under Allen's management.
John Akinde headed Curtis Weston's cross into the bottom corner to put the Bees in front just before half-time.
The hosts secured the three points they thought was required with second-half goals from Richard Brindley and Alex Nicholls.
The draw at the Ricoh Stadium meant the Londoner's needed to win by at least seven goals to save themselves.
Allen, 52, took over from Graham Westley who was sacked in March after a disastrous two-month stint at the Hive.
He won five of his eight games, but it was not enough as Barnet head back to the National League for the first time since 2015.
With Coventry knowing a point would guarantee them a place in the play-offs, there was little incentive for either side to push, and it was no surprise that the match against Morecambe ended 0-0.
There is still a lot of work to do for the Sky Blues if they are to begin climbing back up the leagues but this marks the first time they have finished in the top six of any division since 1970.
Lincoln clinched the final play-off place with a 1-1 draw against Yeovil. It was a tense afternoon for the Imps, who trailed for a long time after Jordan Green scored a 10th-minute opener for Yeovil.
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