The Swans reached only a third semi-final in top-flight cup competitions in their 100-year history thanks to an 81st-minute own goal from Boro's Seb Hines.
Hines could not clear a Swansea corner as the Premier League side exerted sustained late pressure, and the ball flew off him and past Boro goalkeeper Jason Steele.
It put Swansea into the last four alongside Bradford - memorable conquerors of Arsenal - and Aston Villa, while Chelsea meet Leeds in the remaining quarter-final clash next week.
Middlesbrough gave as good as they got for most of a low-key contest in front of a 15,000 crowd, as both sides lacked a sharp attacking edge and laboured for long periods.
But the Swans eventually added Boro to their list of Capital One scalps this season alongside Liverpool, Barnsley and Crawley, while the visitors were left to reflect on what might have been following their 12th successive away League Cup fixture.