The game finished goalless in normal time, as Rangers substitute Charlie Austin saw a goal ruled out for offside in the final ten minutes.
The match then went straight to penalties, as Aiden McGeady, Ross Stewart and Alex Pritchard converted from the spot.
QPR defender Yoann Barbet missed the decisive penalty as he blazed his effort off target to send Sunderland through.
The result led to big celebrations in the away end as 2,731 supporters packed out the away end at the Kiyan Prince Foundation Stadium.
Johnson’s side will now wait for Wednesday’s draw to see who they meet in the last eight of the competition.
Our chief Sunderland writer Phil Smith has delivered his verdict on how each player fared for the Black Cats.
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1. Lee Burge - 9
Had an excellent night. Made some good saves, for the most distributed well and cleared some dangerous set pieces. Started the shootout with a fine save that proved key. 9 Photo: JPI Media
2. Carl Winchester - 7
A real test, given the step up in opposition. That meant he was allowed less time on the ball and fewer options to hit, but he came through the challenge well enough. Defensively stood up well. 7 Photo: JPI Media
3. Frederik Alves - 7
Had one let off when slicing a clearance onto the roof of his own net, but for the most part very composed and did his defensive work well. Looked comfortable against Championship opposition. 7 Photo: JPI Media
4. Bailey Wright - 8
Showed his Championship experience. Won a lot of his duels and generally swept up in dangerous situations very well. One or two uncertain moments in possession but defensively robust. 8 Photo: JPI Media