It was cruel on Vale's League Two visitors who had been the better side and created most of the chances in a game which came to life in the second half.
A first-minute shot from Ben Williamson forced a decent save from Bury debutant Jensen, but there was little else to warm the crowd during a dull first half.
The visitors took a deserved lead in the 71st minute after Temitope Obadeyi showed Vale right-back Richard Duffy a clean pair of heels.
Obadeyi's low cut-back from the left byline was sliced by Sinnott, but the ball flew straight to Chris Sedgwick who stooped to head home inside Neal's right-hand post from six yards.
Vale levelled in the 79th minute when substitute Chris Shuker's right-wing corner was headed in by Pope from five yards.
But it took three fantastic saves by Neal, all to deny Bury striker Shaun Harrad, before Pope settled it with just seconds remaining.
- Sports & Recreation
- Chris Neal