An excellent strike from Dan Neil put the hosts ahead in the 15th minute but Stanley defender Michael Nottingham equalised before half-time.
Sunderland right-back Carl Winchester then scored the winning goal in the 58th minute after converting Aiden McGeady’s pass with a neat finish.
The hosts had chances to kill the game off but Ross Stewart and Elliot Embleton couldn’t secure a two-goal cushion.
Johnson’s side defended admirably, though, with new goalkeeper Throben Hoffman making his first start for the club between the sticks.
The results means the Black Cats are now two points clear at the top of the table.
Our Sunderland writer Phil Smith has delivered his verdict on how each player performed at the Stadium of Light:
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1. Thorben Hoffmann - 5
A tricky introduction to league football. Little he could do about the goal and his distribution was largely sound, but unconvincing from set plays. Will be better for this challenging debut against a side with plenty of aerial threat. 5 Photo: JPI Media
2. Carl Winchester - 7
Winchester’s running power has been impressive this season and yielded another reward with that fine overlap to seal the winner. It was a nerveless finish, and another strong performance. Fast becoming a cult hero. 7 Photo: JPI Media
3. Tom Flanagan - 6
Faced his toughest test of the season against an imposing forward line, but came through it and looked strong in the second half. 6 Photo: JPI Media
4. Callum Doyle - 6
One or two dangerous moments in possession that almost cost his side but came through a tough battle well and showed his class at times. 6 Photo: JPI Media