N'Golo Kante fouled Tom Carroll deep in midfield, and the Swansea key midfielder Gylfi Sigurdsson stepped up to take the free-kick.
Middlesbrough are without a win in nine league games and lie one point above the bottom three.
Cesc Fabregas has insisted that he wants to stay at Chelsea and fight for his place despite falling out of favour under Antonio Conte. His goal was well taken and reminiscent of Frank Lampard in how he arrived late into the box, but it was the way he kept Chelsea ticking over that was most impressive and he was central to all of their best moves.
Chelsea started on the front foot and in the third minute Swansea keeper Lukasz Fabianski was forced to palm out a fierce shot from Costa after he was picked out by Pedro.
Swansea's defensive players also got a lot of stick from him soon after, as Pedro completed a superb solo run from wide on the left all the way into the area, where his shot was eventually blocked for a corner.
Diego Costa wrapped up the points when he finished off Eden Hazard's pass in the 84th minute for his 17th goal of the season. I understand his frustration and the situation but I will call him, which is natural after this - to talk and to show my disappointment for this decision as I think it is very strong.
"He has always had this quality but the formation we are playing is now better exploiting his characteristics".
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The Swans face a Chelsea side that now sits eight points clear at the top of the Premier League table.
In-form defenders Alfie Mawson and Martin Olsson scored the goals that dispatched the champions, and have shored up the defence to the point where they are among the first names on the team sheet.
Antonio Conte's side have been imperious during that run, conceding just five times - two of them in the same game against Stoke on New Year's Eve.
Referee Neil Swarbrick rejected Swansea's appeals and Clement later said: "I thought it was a penalty". "Swansea", though he added defensively, and so many goals will be. I've done a lot in the last six years, working in three different countries, learning different languages, learning different football cultures.
Conte also made sure to praise the contribution of Pedro, who was selected over the Brazilian Willian.
"But we won one of those games and were unlucky not to get a positive result in the other, and I think both those teams would say we gave them hard games". However, the Swans were able to pull a goal back through Fernando Llorente from a set piece on the stroke of half time.