Michael Laudrup’s side scrape past their Championship opposition to reach the semi-finals, although they were slightly fortunate to get the better of Tony Mowbray’s team
Swans boss Michael Laudrup made a number of changes from the side which lost to Norwich City in the Premier League on Saturday, most notably bringing in Itay Shechter for Danny Graham.
Elsewhere free-scoring Spaniard Michu got another chance to add to his goal tally of 14 goals in all competitions so far this campaign.
As for the visitors, Boro boss Tony Mowbray named an attacking line-up as he looked to secure his side's passage into the semi-finals of the competition.
Both teams started the game solidly, passing the ball around well but ultimately failed to create any clear-cut opportunities.
The only chance of note in the opening 20 minutes fell to the visitors and ex-Everton forward Lukas Jutkiewicz, but his poor control resulted in him squandering the opportunity to test Gerhard Tremmel in goal.
However, the hosts' German keeper was called in to action just a minute later when former Sunderland midfielder Grant Leadbitter forced him into comfortable save with his effort from just outside the box.
Middlesbrough continued to prove more than a match for their Premier League opponents but came close to falling behind just past the half hour mark, when Shechter guided his header on target but his effort lacked power.
Despite scoring freely over recent weeks, Swansea struggled to create any real chances and both set of players headed into the half-time interval with the match still goalless.
The second half opened with both teams seemingly intent on scoring a long-range effort but like much of the match so far, all of the attempts lacked any real quality.
Both managers looked to their benches with a third of the match left as Laudrup brought on former Celtic man Ki Sung-Yueng while his counterpart opted for Andy Halliday and Richard Smallwood.
The best chance of the match arrived in the 74th minute when on loan Villareal midfielder Jonathan De Guzman forced Jason Steele into a full length save from a free kick to keep the game goalless.
However, the highly-rated goalkeeper was unable to stop his side from falling behind eight minutes later. De Guzman was again in the thick of the action, this time playing in a corner which was put into his own net by Hines.
As much as the visitors tried to push for a goal to take the match into extra time they failed to create any real chances before the referee blew for full-time, and will be hoping to get back to winning ways in the Championship at the weekend.
Swansea's victory see’s them take their places in the semi-finals of the Capital One Cup alongside Aston Villa and giant-killers Bradford City who dramatically beat Arsenal on penalties, with Leeds United and Chelsea to face each other for the final spot.