Debutant Steve Basham kept his nerve to convert an injury-time penalty and earn Hayes & Yeading a share of the points.
The on-loan Luton striker came off the bench to drill home a low shot after Gary Elphick was ruled to have handled in the box.
Jamie Taylor's rasping 20-yard strike saw Eastbourne Borough take the lead after nine minutes.
Basham also saw an angled shot clear the bar while Taylor missed the chance for a second goal by heading wide.
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Eastbourne Borough manager Gary Wilson told BBC Sussex:
"I'm not a moaner or a groaner but when blatant decisions like that go against you it's shocking, and I'm really disappointed in it.
"Their player's pushed Gary Elphick right onto the ball, and it's hit Gary's elbow, but he's been pushed blatantly onto it.
"How he can give them a penalty for that, I don't understand it."
Hayes & Yeading: Overland, Mulley, Cadmore, Ruby, Green, James (Fitzgerald 82), Marc Canham, Marwa, Binns, Sean Canham (Basham 62), Watkins. Subs Not Used: El-Abd, Little, Baker.
Goals: Basham 90 pen.
Eastbourne Borough: Knowles, Baker, Austin, Elphick, Brown, Taylor, Armstrong, Weatherstone (Johnson 57), Matt Crabb, Enver-Marum, Benjamin. Subs Not Used: Atkin, Jordan, Rowe, McLaggon.
Booked: Brown.
Goals: Taylor 9.
Att: 349.
Ref: A Bennett (Devon).
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