The hard-hitting southpaw caught the champion with a swinging left hook with less than 90 seconds of the contest gone and, although Dawson tottered to his feet, referee Michael Griffin spared him further punishment.
It was Dawson' second successive defeat - and a third career loss - after he had dropped down to super-middleweight for an unsuccessful shot at Andre Ward's title.
Stevenson was also coming up from 168lbs, and the 35-year-old's victory had the WBC's No 1 contender Tony Bellew licking his lips in the Sky Sports studios.
Liverpudlian Bellew, who settled his argument with Isaac Chilemba recently at the second time of asking, said: "Adonis Stevenson is wild and he gives you opportunities, so I don't think he's going to be a long-standing champion - especially if I've got anything to do with it.
"I'm not going to have to go looking for him, but at the same time he's not going to have to go looking for me. I will be there to meet him, I ain't going to run from nobody.
"I've got myself into a position I wanted, it's always been a dream of mine to fight for that belt and it's here now.
"It doesn't matter where this fight happens - his back yard, England, Liverpool...it's going to be a fight to watch and I'm so excited by the prospect of facing Adonis Stevenson."