For the second time in five days City played the team in second place and as on Tuesday when Fylde went to the top after their 0-4 win, this time it was Barrow who went top after a 0-2 Aggborough win.
Goals from Anthony Wilson and substitute Andy Cook earned Barrow the points after City squandered an early chance to go into the lead with a missed first half penalty.
Barrow had early chances but both sailed high over the bar from Andy Howart and Anthony Wilson. City were well in the game when on 24 minutes Andy Burns brought Sean Geddes down in the box giving City a penalty. However, The Bluebirds keeper guessed the right way to save Geddes's spot kick.
Ten minutes before the break it was Barrow who grabbed a goal on the counter attack. Anthony Wilson showed great skill to control a Danny Livesy ball just inside the box before firing past Ryan Boot.
The second half saw City with an early opportunity when from a cross Taylor kept his side in the lead once more by saving a point blank range header from Daniel Nti, with superb reflexes to tip over the bar.
Barrow had more opportunities, the best of which was from Jason Walker who took down a Sam Sheridan flick on his check before unleashing a volley which went just over.
With 15 minutes remaining City fans thought an equaliser was coming when a free kick was awarded 25 yards out but Taylor was not troubled from the Sean Geddes goal bound shot.
In the third minute of added time, Barrow sealed the win when their leading goalscorer Andy Cook raced goal bound, Ryan Boot saved the initial shot but Cook was first to react and tapped the rebound into the goal to make it 0-2 to send the travelling fans home happy after hearing second place Fylde could only manager a 1-1 draw at home to Hyde.
One bonus for City was that Tyler Weir returned from injury with 15 minutes to go.