A first triumph over the Swans on Teesside in 33 years, and a first victory of any sort over their opponents since a Championship clash in 2007, lifts Boro into 14th place in the table - now six points ahead of Swansea, who are only kept off the bottom by Hull City.
The visitors made a bright start to the contest, testing Victor Valdes early on through a well-struck Gylfi Sigurdsson shot, but against the run of play Boro scored from their first real attacking move.
Despite the ball being slightly behind him, Negredo met it sweetly on the half-volley and sent it past the reach of Fabianski, taking his tally for the season to four goals in 16 appearances.
That goal ensured that Swansea have now gone behind in 12 of their 17 fixtures so far this season, while also making Negredo Boro's outright top-scorer in all competitions this term.
City boss Bob Bradley was unhappy on the sideline, claiming that the ball had already gone out of play before Jay Fulton had touched it out for an opposition throw in the immediate build-up to the incident that led to the Smoggies' killer second.
Gaston Ramirez, back in the side after passing a late fitness test, also skimmed the roof of the net with his free kick, but that proved to be it in terms of first-half chances.
The second half provided a carbon copy of the first as Swansea again enjoyed more of the possession but soon conceded, this time after De Roon smartly tucked home at the end of a free-flowing Boro move.
Stewart Downing, brought on for the injured Viktor Fischer prior to the Smoggies' opening goal of the match, saw an attempt of his own kept out down the middle and Ben Gibson also headed narrowly over the bar.
Boro, the division's lowest scorers prior to today's fixtures - failing to register in three of their last five - had done the hard work before the game had reached its final quarter and saw out the remainder with ease.
Leroy Fer did have one effort blocked aside in front of goal, while the returning George Friend blasted a shot wide when slipped through by Fabio Da Silva up the other end, but the final act belonged to Valdes who superbly kept out a close range header from Fernando Llorente to keep his side's clean sheet intact. body check tags ::