Hayes and Yeading moved to within a point of safety with a hard-earned Blue Square Bet Premier victory at Bath.
Michael Thalassitis nodded in the winner when he latched onto keeper Steve Arnold's long clearance.
Thalassitis then headed over just after the break, while Julian Owusu's long-range effort failed to trouble Jason Matthews in the home goal.
The hosts pushed for an equaliser but Scott Murray and Chris Shephard were both thwarted by Arnold.
VIEWS FROM THE DRESSING ROOMS
Bath City manager Adie Britton told BBC Bristol:
"We've lost the game, as with so many games this season, due to one poor mistake.
"It has been a difficult season. You can do two things; you can lie down and give up or you can continue to fight.
"We will continue to fight and we will continue to try to impove the side.
"For me, a side that has fallen apart is a side that doesn't have any hope, doesn't play any football, doesn't have any spirit. I didn't see a team that had fallen apart tonight."