Ba’s 68th minute strike proved the difference, although Swansea had to play all but 16 minutes with 10 men after Chico Flores was sent off for two yellow cards in as many minutes.
Chelsea manager Jose Mourinho decided against attending the post match press conference, but Swansea caretaker head coach Garry Monk was far from impressed by what he described as a “strange” decision to send off Flores.
“If you go by the letter of the law it was a sending off. But if you go by the letter of the law you will have six or seven sendings off every game,” said Monk.
“However, their bench surrounded the fourth official and the players surrounded the ref and the next thing, he has given the red card. I am not doubting Phil (Dowd) at all, it was just strange that it took so long.”
Chelsea assistant manager Steve Holland said; “The two yellow card decisions were crystal clear. I think the referee was correct on both counts. As to who he speaks to and how long he takes, you will have to ask him how it works. It is the fourth time I have been to this stadium and the first time I have been on the winning side. They are a difficult side to build pressure against because they like to keep the ball. On the back of that and after a difficult Champions League game, you worry about your next fixture.
“However, we are happy to get over the line. Demba deserves it. He is always working in training afterwards on his own and he deserves to have the week he has had. All credit to him.”
Having seen Brendan Rodgers’ side kick off the day with a significant victory over Manchester City, Chelsea knew that nothing less than three points at the Liberty Stadium would do.
Mourinho made five changes to the side that beat Paris St Germain on Wednesday night while Swansea, once again, without Michu, left Jonathan de Guzman on the bench but went with an attacking line up that included Nathan Dyer, Wayne Routledge and Pablo Hernandez.
Though the hosts began well, the game turned on the two yellow cards and subsequent dismissal of Flores. Dowd dished out the first for a foul on Willian and the second was brandished for a clinical trip on Andre Schurrle. It took Mr Dowd an age to make his decision with Swansea clearly concerned that the official had come under intense pressure from the Chelsea players.
With a numerical advantage, Chelsea dominated the ball thereafter with Willian becoming increasingly influential. However, Swanseadefended with great strength and desire and the visitors were restricted to one or two chances. Willian miscued from distance and Mohamed Salah, inexplicably fired over from a dozen yards.
Mourinho swapped Schurrle and Ramires for Eto’o and Oscar at half time and they immediately looked more of a potent force. Eto’o was close to converting a cross from Willian, four minutes into the second half before Oscar’s ambitious strike was blocked by Ashley Williams.
Needing to gamble, Mourinho changed his formation dramatically with Eto’o and Ba leading a four-man front line that now included Willian and Salah.
Chelsea continued to knock at the door and somehow, with only Michel Vorm to beat, Eto’o tugged his shot wide. Almost immediately, Swansea broke and with Wayne Routledge outpacing Ivanovic, Swansea were denied the opening goal thanks to a somewhat fortuitous block from Terry. It certainly gave the hosts greater belief and though Chelsea continued to pour forward, they continued to frustrate the visitors with a succession of last-ditch clearances, most notably from captain Ashley Williams and Angel Rangel.
However, the pressure finally told in the 68th minute when Nemanja Matic picked out Ba 30 yards from goal and though Williams was in close proximity, the former Newcastle striker beat Vorm from 20 yards.
SWANSEA CITY (4-2-3-1) M Vorm 5; A Rangel 5, C Flores 3, A Williams 5, B Davies 6; Britton 6 (J De Guzman 85,5), Shelvey 6; Dyer 6 (Ngog 73,5), Hernandez 5 (Amat 16,6), Routledge 6; Bony 6
CHELSEA (4-2-3-1) Cech 6; Ivanovic 6, Terry 7, Cahill 7, Azpulicueta 7; Ramires 6 (Oscar 45, 7), Matic 7, Salah 6, Willian 7, Schurrle 6 (Eto’o 45,6); Ba 7 (Mikel 78,6)
Referee: P Dowd.