In a dramatic finish to an entertaining tie at Borough Park, Wilson grabbed the winner in the 89th minute.
A minute earlier Sean Law had levelled for the County League side, making amends for a penalty miss earlier in the half.
Both goals came from free-kicks just outside the respective areas which were placed perfectly into the top corner.
Wilson had put Workington ahead after 52 minutes with a superb strike from 25 yards.
The Reds, who beat Aspatria to win the trophy in May, dominated the first-half and went close through Wilson, Nathan Waterston and Robbie Hebson.
Aspatria were more of a threat in the second half and had their chance to level on the hour but goalkeeper Tom Fowler went the right way to save Law’s spot kick.
Reds manager Derek Townsley said afterwards: “I was disappointed that we didn’t win more comfortably, because we had enough opportunities.
“We have to start being more clinical in front of goal.
“Rob did score two quality goals for us and it was great to see Tom make the penalty save, but there weren’t many pluses for us.”
Club captain Gari Rowntree was given an hour as he came back from a lengthy lay-off following a knee operation.
That was countered by an injury to Sam Smith in the first half as he was helped off after going over on his ankle.
Reds go to Whitby Town on Saturday in the FA Trophy.
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