
The game was played in front of a bumper crowd with neither side able to find a way through, with Town man Jake Hackett missing a penalty and striker Jacob Hazel forcing two fine saves from Theo Richardson in the visitors' goal.
Richardson's opposite number, Shane Bland, was barely troubled against the league leaders with catches being the most be ever really had to do.
Town were able to close the gap to the play-off spots down to two points, but with fifth-placed Warrington Town losing 3-1 to Ashton United, the opportunity to go level on points was there for the taking.
Lee Bullock and Nathan Haslam named an unchanged starting eleven from the one that defeated Witton Albion 2-0 the previous week, whilst Jameel Ible and Malik Dijksteel returned to the bench.
The Seasiders should have taken the lead in the 26th minute when Josh MacDonald was brought down in the box, but Hackett was unable to hit the target from twelve yards, drilling his spot-kick over the bar.
Whitby kept the pressure up with Ben Turner having to head MacDonald's cross over the bar with Hazel lurking behind him, and Richardson then preventing Hazel from scoring just before the break with an excellent close-range save.
In the second-half, Whitby continued their push for a breakthrough as Bradley Fewster fired an effort over the bar. After neat build-up play from the Blues, a cross from the left-wing found Fewster but he was unable to hit the target on the turn.
Buxton had a half chance in the 75th minute, substitute Nathan Fox dragging wide from just inside the Blues' penalty area, before MacDonald's cross at the other end was crying out for a touch but no one in a Whitby shirt was able to get onto the end of his pacey ball into the box.
Despite being under pressure for most of the game, the last big chance fell to the away side but Dan Cowan was unable to hit the target from a tight angle in the 89th minute.
Whitby Town return to league action on Saturday away at Grantham Town. The clash will take place at the South Kesteven Sports Stadium, with kick-off at 3pm.