However, the leaders were left frustrated after being held to a 1-1 draw, so leading the Ammies by two points.
The home side saw Joe Leesley miss a penalty early on.
But Jack Emmett’s goal on the half-hour mark saw Town take a 1-0 lead into half-time.
However, Avenue pegged the hosts back 5 minutes after the restart though as Adam Boyes headed in Nicky Clee’s cross.
Town poured men forward desperate for a win, but it was Bradford who almost snatched all three points at the death.
The only other Tuesday game saw in-form Gainsborough Trinity make it six games unbeaten in all competitions following a 3-2 Northolme win over AFC Telford United to move three points clear of the drop zone.
Michael Jacklin gave Trinity a 9th minute lead.
But the Bucks took a 2-1 lead into the break, thanks to goals from Elliot Newby after 34 minutes and Jordan Lussey right on the whistle.
Bradley Wells headed in a Josh Lacey throw 5 minutes into the second half to pull Trinity level before Jacklin tapped in the winner just 6 minutes later from a Matty Taylor free-kick.