In an enthralling first half, the hosts took the lead after just three minutes when Josh McQuoid prodded home from the centre of the Barnsley box.
The visitors found themselves back on level terms after Conor Hourihane rifled in a standout effort into the top left corner.
City thought that they had stolen a lead to take into the interval when Jordan Clarke made no mistake from close range in the 44th minute, but Barnsley's Leroy Lita had other ideas as he converted an athletic header just seconds before the half-time whistle.
Clarke did his best to push City forward early in the second half, as he was again denied by the in-form Davies, who was on hand to deny the same player 10 minutes later in the centre of goal.
Sam Winnall attempted to lift an unenterprising Barnsley when he tried his luck from distance, only to see the shot fly inches past the top corner.
The visitors might have won it with just over five minutes remaining, had Reece Brown done a little better with his head, having guided his attempt wide of the left post.