Having scored the goal that sent Jose Mourinho's side into the UEFA Champions League semi-finals against Paris Saint-Germain on Tuesday, the Senegal international - linked with an exit from the club at the end of the season by one Sunday newspaper - pulled the Londoners to within two points of leaders Liverpool with a 68th-minute strike.
With Liverpool having claimed a thrilling 3-2 victory over Manchester City at Anfield earlier in the day, Mourinho's men needed a win to keep in touch at the top of the Premier League.
Their cause was helped when Swansea's Chico Flores was sent off in the 16th minute after receiving two bookings in quick succession, but chances were few and far between thereafter.
However, Chelsea finally found the net through Ba midway through the second half, as he gave the visitors their first win at the Liberty Stadium and left Swansea just three points clear of the relegation zone.
Ba was rewarded for his midweek heroics as a substitute with a place in Mourinho’s starting XI, while Michu dropped out of the Swansea squad with soreness in the ankle that kept him sidelined for three months earlier in the season.
His replacement, Pablo Hernandez, was first to produce a shot at goal – forcing Petr Cech into a diving save with a low drive from 20 yards in the fourth minute.
Mohamed Salah should have put Chelsea in front two minutes later, but the Egyptian side-footed wide of the left-hand post after good work by Branislav Ivanovic down the right.
Swansea looked bright in attack and Wilfried Bony had Cech at full stretch with a superb header 14 minutes in.
The home side were dealt a huge blow when centre-back Chico received two yellow cards in the space of two minutes – the Spaniard dismissed for the second time in just over a month after his sending off against Crystal Palace.
Chelsea began to enjoy the better of the possession having gained a numerical advantage, but with the hosts sacrificing attacker Hernandez to bring on defender Jordi Amat, the Swansea back four continued to hold firm.
Nathan Dyer stung the palms of Cech shortly before the break, but head coach Garry Monk will have simply been relieved to reach half-time all square.
Chances were at a premium in the early stages of the second half, but Ba glanced a header across the face of goal in the 55th minute from a dangerous Ivanovic cross, before Samuel Eto'o skewed an effort wide from close range.
Wayne Routledge's shot then appeared to deflect behind off the arm of John Terry, though the incident was met with few calls for a penalty, before Ba dealt Swansea a crucial blow.
Nemanja Matic drove ball into the path of Ba, who rounded Ashley Williams and sent the ball beyond goalkeeper Michel Vorm, who may feel he could have done better with a shot that squirmed under his body.
Chelsea continued to push in the closing stages and a moment of madness from Vorm almost gifted them a second late on when he ran out of his area to challenge Eto'o - but the striker's shot was blocked.
