Heavy rain throughout the day made the pitch unplayable at Bootham Crescent, despite the best efforts of ground staff.
Kick-off - initially scheduled for 19.30BST - was delayed for half an hour, but referee Carl Boyeson decided to postpone the match before the rearranged start time of 20.00.
The first leg will now be played on Monday, barring any further issues.
York finished the League Two season in seventh after an excellent 17-match unbeaten run to end the campaign.
Fleetwood were five points ahead of them in fourth, missing out on automatic promotion by just five points.
The two sides - who are both battling for a place in the third tier of English football - shared the honours in their regular season meetings.
Fleetwood won 2-0 at York in October before Nigel Worthington's side won the return fixture 2-1 in February thanks to Wes Fletcher's last-gasp winner.
