Swansea sink Boro to reach League Cup semis
The Middlesbrough defender headed a Swansea corner into his own net in the 81st minute after an evenly matched quarter-final at the Liberty Stadium in Wales.
Middlesbrough enjoyed the best of the first-half chances and came close to breaking the deadlock when Grant Leadbitter side-footed a shot that had Swansea goalkeeper Gerhard Tremmel scrambling to save.
Swansea upped the pressure after the break and Jonathan de Guzman's free-kick was tipped on to the bar by Middlesbrough keeper Jason Steele before Hines headed into his own net.
"It was a very important victory and now we're in the last four," Swansea manager Michael Laudrup said.
"They are a good side, and we may be playing them next year in the Premier League. In the second half we came out and dominated like we normally do at home."
Fourth-tier Bradford City upset Premier League Arsenal on Tuesday to secure a semi-final spot alongside Aston Villa who beat Norwich City.
Chelsea, who are playing in the World Club Cup in Japan, face Leeds United in the final last-eight clash on December 19.