With exactly 100 places separating the two sides in the English league pyramid, League Two side Exeter were overwhelming favourites to win the first-round clash at Cantilever Park.
However, Warrington - who are currently 17th in the Northern Premier League Division One North - progressed courtesy of Craig Robinson's seventh-minute goal.
Robinson beat his marker at the near post from David Mannix's corner and flicked home a header that Exeter goalkeeper Christy Pym failed to keep out.
Following the victory, Warrington's players were flanked by the club's supporters, who invaded the pitch in jubilant celebrations.
"I can't believe it. I am speechless. This is what I have dreamt about," Robinson told BBC Sport after the game.
"I am so happy my family are here and I am so happy for the club."
Robinson also revealed that he was struggling in the build-up to the game with injury.
"I have been struggling with an injury all week but there was no way I was going to miss it," he added.
Warrington manager Shaun Reid, the younger brother of former England international Peter, said: "These boys deserve it. No one will want to come here in the next round. Everyone was a hero. They died for me."
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