Jordan Hulme made it 3-0 with a fine solo effort on the hour mark, but Lawds then pulled one back.
But any prospect of a nervy finishe ended when substitute James Rosenburg rose highest to head home
Flixton can now look forward to a semi-final showdown with top-flight side Saints.
Delmonte told JC Sport: "It was a good performance and we never looked like losing.
"The goals in the first-half gave us some breathing space to enjoy ourselves after the break.
"We now move on in this competition – and there is no side that we will fear."
Report by Darren Witcoop