Mark Roberts volleyed Stevenage ahead from close range and David Bridges doubled the lead with a looping header.
Beardsley fired home a third from long-range before Cole's late heroics.
Emphatic as the final scoreline was, Eastbourne matched their promotion-chasing opponents for the first half-hour but will rue a sloppy start to the second period and their disastrous late capitulation.
Jamie Taylor had the hosts' best chance with a shot that clipped the bar with the scores at 0-0.
And Taylor also twice narrowly failed to connect with teasing crosses.
But, having edged ahead when a clever free-kick saw Joel Byrom and Scott Laird combine to tee up Roberts, the visitors took control two minutes after the interval.
This time it was Bridges who benefited from some slack marking from another Byrom set-piece, the striker nodding past keeper Danny Knowles.
At 2-0 Eastbourne's resistance was broken but it got much worse for the home side.
Beardsley rifled in a spectacular third from 25 yards after some more poor defending.
And Cole, who only came on after 72 minutes, piled on the misery with a quick-fire treble in the final nine minutes.
His first was smashed into the roof of the net from the spot after Beardsley was felled and two tidy finishes from six yards wrapped up a victory which cuts Oxford's lead to a point following their draw with Cambridge.
Eastbourne Borough: Knowles, Austin, Baker, Pullan, Elphick (Atkin 74), Taylor, Benjamin, Johnson (Smart 64), Matt Crabb (Jenkins 64), Armstrong, Enver-Marum. Subs Not Used: Weatherstone, Nathan Crabb.
Sent Off: Baker (82).
Stevenage: Day, Ashton, Laird, Albrighton, Roberts, Vincenti (Long 63), Byrom (Odhiambo 84), Bridges, Bostwick, Beardsley, Odubade (Cole 72). Subs Not Used: Bayes, Maamria.
Goals: Roberts 40, Bridges 47, Beardsley 71, Cole 83 pen, 86, 90.
Ref: Nick Kinseley (Essex).