Adam Johnson fired Boro ahead before goals from Marvin Emnes and Tuncay secured a first league success on the road in nine months for Gareth Southgate's men.
Highly-rated England under-21 star Johnson opened the scoring with a low rasping shot before Emnes unleashed a thunderous 25-yard drive and Tuncay headed home.
Boro could have scored more but centre-back David Wheater powered Johnson's right-wing corner against the crossbar in the sixth minute.
The hosts, under new manager Paulo Sousa, rallied and dominated possession and territory in the early exchanges against Southgate's fallen Premier League outfit.
Both sides were creating chances aplenty as Stephen Dobbie warmed the palms of Danny Coyne with a long-range shot before Leroy Lita and Mark Yeates fired over for the visitors.
Lita, making his first league start for the Teesiders following his move from Reading on a free transfer, found space at the far post but could not keep his header down from six yards.
A lively yet fiesty contest at the Liberty Stadium saw Boro midfield pair Rhys Williams and Gary O'Neil booked by referee Paul Taylor for late challenges.
But Boro settled quickly to their task and got their just rewards on 32 minutes when Johnson fizzed a cross-cum-shot past Dorus de Vries into the far corner.
Swansea nearly drew level two minutes after the re-start but Dobbie failed to find his range from Angel Rangel's cut-back.
But Dutch hitman Emnes, who struggled to adapt to English football last season, capitalised after Ashley Williams' mistake and his long-range strike stunned Swansea goalkeeper de Vries.
Nathan Dyer saw an effort flash wide for the hosts but Boro were in full control and finished the game stronger as Johnson and substitute Tuncay went agonisingly close.
But the Turkish forward, who has been linked with a move away from the Riverside, was not to be denied when he nodded home Johnson's pin-point corner with nine minutes remaining to seal victory.