Goals during the early and latter stages of the first half sent Royal Albert tumbling out of the Qualifying Cup.
Lowland League outfit Spartans opened the scoring in the fourth minute when Dean Horribine found the net with a header.
With two minutes of the first period remaining Spartans went further ahead.
On being fed by an Aaron Murrell pass, Keith Murray tucked a shot past Hawick keeper Kyle Rankin.
Twenty minutes from time, goal scorer Murray was red carded for a last man challenge.
Commenting on the game, Hawick team boss Dean Shanks said: “Although we lost I was pleased with our performance.
Spartans are a good Lowland League side and I think a lot of people thought Hawick would get a hammering but we gave them a game.”