Stevenson, 35, finished Dawson with a left hook after a minute dead and he was counted out to leave Stevenson dancing around the ring and whooping with joy.
He said ringside: "You understand why people don’t want to fight me now?
"The game plan: use my speed, put the pressure on and, as soon as I had a good chance, put my left at the top.
"I put my left, I caught him, that’s a beautiful punch."
Stevenson improves to 21-1-0 (19 KOs). Liverpudlian Tony Bellew is in line to face him next as the mandatory challenger to the WBC belt having won the final eliminator against Malawi's Isaac Chilemba in May.
American Dawson, 30, who also lost his last fight against Andre Ward, is now 31-3-0 (17 KOs).
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