The only goal of the game came with nine minutes remaining as Seb Hines headed a Jonathan De Guzman corner past Jason Steele and into his own net.
It put Swansea into the last four of the competition for the first time in their history alongside Bradford - memorable conquerors of Arsenal on Tuesday night - 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.
Grant Leadbitter went close for Boro, League Cup winners eight years ago, with a 21st-minute effort that forced Swans goalkeeper Gerhard Tremmel into a save low at his right post.
Tremmel was then forced to react smartly to keep out a Leadbitter effort, and Boro boss Tony Mowbray was the far happier manager at half-time in a tie that never seriously threatened to move out of second gear.
It was no surprise when Michael Laudrup made his first change just 10 minutes after the break when he sent on Luke Moore to replace Itay Shechter. Moore was immediately in the action, challenging Boro defender Andre Bikey just outside the visitors' penalty area that ended with both players going down.
Bikey required treatment before he could continue, but even though he attempted to run off his injury, it was to no avail. Faris Haroun also departed for Boro, with Andy Halliday and Richard Smallwood going on.
Laudrup also made a second switch, with Ki Sung-yueng going on for Wayne Routledge, but there was still no sign of the deadlock being broken.
Swansea, though, started to string some attacking phases together, and they went agonisingly close 15 minutes from time after Nicky Bailey was cautioned for bringing down Nathan Dyer. De Guzman took the resulting free-kick, and Steele palmed it on to the crossbar, keeping Boro level, although pressure continued to mount with Michu a consistent threat.
Referee Lee Probert then waved away Swansea claims for a penalty when Hines challenged Ki, and a game that had previously lacked an attacking flow suddenly opened up.
Boro could only survive such sustained pressure for so long, and the unfortunate Hines diverted the ball beyond Steele from a corner under pressure from Garry Monk as Swansea went ahead to ultimately secure a last-four place.