The Minstermen were unlucky not to open the scoring after seven minutes with Michael Coulson seeing his shot blocked by Lee Bell after rounding goalkeeper Rob Lainton in the area.
Burton had started the stronger out of the two sides and twice went close to taking the lead with Nick Pope saving well from Callum McFadzean and repelling Bell's 25-yard stinging drive too.
The visitors were to go even closer just before half-time with Billy Kee controlling the ball on his chest 25 yards from goal before firing against the crossbar.
York had Pope to thank on the hour mark for keeping the scores level as he raced back to his line to gather the ball when it looked like Matty Palmer's effort would be trickling into the net.
There was to be no breakthrough in the game, though, with Keith Lowe heading narrowly wide for the hosts in the closing stages.