The triumph follows last year's ten-wicket win over Zimbabwe and batsman Jason Molins, who top scored in Ireland's run chase with 58, believes the Irish amateurs have made their case for entry to the English One-day National League.
"I think we are more than capable, as we proved here, and beating Surrey certainly won't harm our credentials," Molins said.
"This win can only raise our profile and the third round game should attract lots of publicity.
"There are lots of things we compete against like football, rugby and Gaelic sports but the interest is there.
"The wins against Zimbabwe and Surrey are great, but they won't get us to the World Cup - that is the priority."
Molins insists he was always confident of victory after restricting Surrey to just 261 during a rain-affected first day.
He added: "Obviously for a side like ourselves to come up against a full-time side of professionals like Surrey and beat them as comprehensively as we did is massive.
"It is hard to compare this win to the one over Zimbabwe because they were under different circumstances.
"Both are great achievements considering we are amateur and they are both full-time.
"There are eight internationals in their team - six English and two Pakistani. For us to come up against that level and get a result is amazing."