Simon Weaver’s Conference North outfit led at the break but Town were pegged back in the second half by the Ryman League Premier Division side.
Harrogate’s Paul Beesley spurned a super chance on 18 minutes after being sent clear by Dave Merris.
After Beesley just failed to get on end of a White header, White produced a saving block to deny Zac Attwood.
Harrogate took the lead on 41 minutes. Dave Merris’s corner was hooked across goal by White and nodded in by Tom Platt.
But in the 61st minute, the side from a league below levelled, with Jamie Crellin equalising for Hastings.
It was end-to-end in the closing minutes.
Attwood missed a great chance to win it for Hastings when he shot tamely.
Then Town’s Chilaka headed against the cross bar at the other end.
Despite the draw, Harrogate are still in the hat for the third round and it was a memorbale day for the club with the attendance at Wetherby Road an excellent 2,926.