Striker Tony Gray gave the Yellows a half-time lead and added a second in the second half.
Further goals from Jack Mackreth, with his first for the club and a 90th minute effort by Richard Brodie, saw Town through to Monday’s draw.
It is only the second time Warrington have progressed to the 2nd qualifying Round since their historic 1st Round proper victory over Exeter City during the 2014/15 season, when they eventually bowed out in the 2nd round proper at Gateshead.
Town are back in action in the league midweek with a trip to Hyde United and welcome Marine to Cantilever Park next Saturday, as the fixtures come thick and fast.
Tony Gray opens the scoring – Picture John Hopkins
Tony Gray makes it 2-0 – Picture John Hopkins
Jack Mackreth scores his first goal for the club – Picture John Hopkins
Richard Brodie makes it 4-0 – Picture John Hopkins
Town: Tony McMillan, Adam Carden, Adam Hughes, Mark Roberts (Capt) David Raven, Jack Mackreth, Robbie Evans, Tony Gray, Josh Amis, Dyan Vassallo.
Subs: Richard Brodie, James Caton, Bohan Dixon, Mitch Duggan, Jack Dunn.
Burscough: Maxwell Povey, Joshua Dunroe, Jamie Clarke, Luke Gibson, Cameron White, Daniel Brady (Capt) Gary Grier, Peter Henerty, Terry Cummins, Calvin Ekpiteta, Prince Ekpolo.
Subs: William Doherty, Matthew ward, Alberto Canache, Zaid Bangi, Lee Robinson.