Brigg Town had an early chance when Dayle Hutson found Scott Phillips in space, but he fired over the crossbar.
Soon afterwards Martin Pembleton’s free kick was straight at goalkeeper Connor Eastham for a comfortable save.
Pembleton then came close a couple of times as Brigg enjoyed most of the possession in the opening 20 minutes.
Runcorn scored against the run of play through Oli Scott, who buried the rebound after Marcus Perry’s effort was well saved by Tom Collins.
Brigg responded immediately and equalised just two minutes later when Stephen McCarron found the bottom corner of the net from 20 yards.
After equalizing, Brigg’s skipper made an excellent last-ditch challenge on Chad Whyte, who had raced clear at the other end.
The visitors looked dangerous for the remainder of the first half with debutant Jack Coop forcing Collins into a smart stop.
Brigg seized the initiative from the start of the second half with McCarron shooting wide and were rewarded with a goal from Kevan Kahoussi. The youngster latched onto Pembleton’s through ball and rounded Eastham to give the Zebras a 2-1 lead.
Minutes later Hutson ran onto Collins’ long punt forward before shooting narrowly wide.
Kahoussi was continuing to create problems for the Runcorn defence and set up Brigg’s third goal. After dribbling past Jack Fleming on the right wing he picked out Scott Phillips, whose firm strike found the back of the net.
A repeat of the Kahoussi-Phillips combination almost led to a similar goal, but Eastham denied him on that occasion.
Runcorn kept going late on and Coop dragged a shot wide before Keane Joyce headed Perry’s cross just over. However, Brigg maintained enough control of the game to run out 3-1 winners.
The Zebras have a trip to Derbyshire to play New Mills in the next round.