The Valiants had their pick of chances throughout the first half but steady City defending prevented any lead.
Stephen McPhee began the barrage in the 37th minute when he made headway inside and struck a shot that forced an excellent save from Steve Phillips.
Billy Paynter was the next to try his luck on the away goal and should have capitalised when McPhee crossed from the left to the back post, but the yongster fired his half volley weakly wide of Phillips' post.
City did occasionally break out from under the Vale pressure but their finishing was just as poor with Luke Wilkshire having trouble finding the target. Lee Peacock gifted him a free shot inside the area but the Australian spooned his effort over the bar.
The deadlock was finally broken in the 66th minute after Stephen Brooker beat the offside trap and turned on the left of the area to squeeze a shot inside Phillip's right post.
Their lead was doubled 11 minutes later when after Neil Brisco fed him the ball just over the half way line, Mark Bridge-Wilkinson attacked the Bristol box and reaped the rewards placing the ball past the keeper into his left corner from 15 yards.
City did not lie down and grabbed one back just before the final whistle, however, it was only to be a consolation when Peacock headed home Brian Tinnion's corner from close range.
City boss Danny Wilson said: "I think the whole game hinged on the first goal. Up until then, we had a solid grip on everything.
"It is important that we grab the up and coming games by the scruff of the neck as they are going to come thick and fast."