Vics' boss Steve Burr took a gamble on the fitness of Brayson and it paid off richly with the striker scoring three fine goals to take his tally to nine in the FA Cup alone this season.
Morecambe started the game well enough with Northwich keeper Kris Rogers making two excellent early saves from Jim Bentley and Wayne Curtis.
Sean O'Connor then directed a shot and a header off target before the Vics took the lead with their first real attack of the game.
Johnny Allan beat Jim Bentley to a header and his flick fell neatly for Brayson. The former Cheltenham striker kept his cool to round Ryan Robinson and slotted the ball into the net, despite the best efforts of Chris Blackburn on the Morecambe line.
The visitors then began to take a foothold on the game with Danny Mayman dragging a left-foot shot wide from a great chance, before Jimmy Kelly forced Rogers into another good save from a sharp volley.
Minutes later the Vics made it 2-0. The Morecambe defence failed to clear a left-wing Carlos Roca corner and, as the ball was drilled back into the Morecambe box, Brayson reacted sharply again to drill home his second
The second half started with the Vics on top with Ben Chapman firing into the side netting before Morecambe threatened with Bentley volleying over. Michael Twiss let fly with a free kick that was deflected wide and Michael Stringfellow saw a shot well held by the impressive Rogers.
A Garry Thompson corner was headed over by Darren Kempson as Morecambe pushed forward. As they pressed the gaps appeared, however, and Brayson took full advantage again.
Stringfellow was dispossessed on the halfway line by Stuart Elliot. His ball found Brayson in acres of space on the edge of the box and he curled a delightful right footer into the bottom left-hand corner of Robinson's goal.
Michael Carr almost made it 4-0 with a header against the crossbar before the Shrimps scored a late consolation when Danny Carlton slid home from Twiss's flick.