The Ducks started well in front of a record crowd of 864, and overhead kick by experienced former Harrogate Town and Alfreton Town striker Paul Clayton forced the first serious save from visiting keeper Joel Dixon.
But, after just 9 minutes, Shaw Lane did have the lead when Alex Byrne's corner was headed home at the near post by Spencer Harris.
Byron Harrison had a header go just over the bar at the other end, but it was the Ducks half as they went into the break a goal to the good.
However, just 2 minutes into the second half, the Bluebirds were back on level terms when a foul on Chris Clements earned them a penalty which Harrison duly converted.
The home fans expected Barrow to take control now, but they never really took the game by the scruff of the neck
And with 14 minutes remaining, Shaw Lane took the lead for the second time and again it was courtesy of one of their experienced forwards, this time ex-Altrincham favourite Damian Reeves, who controlled a long throw and hammered a shot past Dixon.
Barrow now had to act to stay in the cup and they went close on a few occasions, especially through sub Adi Yussuf, Clements with a 25-yard volley and Harrison.
And the visitors' misery was complete when they had Jimmy Dunne sent-off for a rash challenge on Godwin Abadaki.