Just a few days after poaching the goal which sent Chelsea into the semi-finals of the UEFA Champions League, the striker's crucial effort in south Wales kept Jose Mourinho's side firmly in the title race after Liverpool had beaten Manchester City 3-2 earlier in the day.
Mohamed Salah side-footed an excellent early chance wide for Chelsea before Swansea defender Chico Flores was sent off on 16 minutes following two quick bookings.
Flores was first cautioned for bringing down Willian as the Brazilian tried to launch a quick counter-attack and was then shown a second yellow card, following lengthy deliberation by referee Phil Dowd, for a trip on Andre Schurrle just outside the penalty area.
Chelsea, though, struggled to take advantage of their numerical superiority for long periods, with half-time substitute Samuel Eto'o flashing an opportunity off target before the breakthrough goal arrived on 68 minutes.
Ba's low shot from the edge of the area packed too much power for Swansea keeper Michel Vorm to ensure Chelsea are still two points behind Liverpool with four games to play, and a decisive meeting at Anfield to come in a fortnight's time.
For Swansea, Garry Monk's side are still sweating on their Premier League status following a run of two wins in 11 games which has left the club just three points above the relegation zone.
A fast opening saw Chelsea goalkeeper Petr Cech save low down from Pablo Hernandez, before Salah missed a golden sixth-minute opportunity for the visitors when he side-footed wide from inside the penalty area.
Swansea enjoyed the lion's share of territory, and Cech was called on again to deflect a Wilfried Bony header over the bar before Flores became an unwanted centre of attention.
He was cautioned for a foul on Willian, then he pulled down Schurrle just outside Swansea's box, and referee Phil Dowd brandished another yellow, leaving the home side down to 10 men.
Swans head coach Monk was left fuming by the decision - Dowd appeared to take a long time before he brandished a card - and he cut an animated figure on the touchline before a sense of order was restored and a reshuffle meant Jordi Amat going on for Hernandez.
It meant Swansea were left with a mountain to climb, but despite their disadvantage in terms of personnel there was an assuredness about the Welsh club, exemplified by skipper Ashley Williams.
Chelsea displayed their trademark patience in attempts to break Swansea down, but keeper Vorm had still to be seriously troubled as the game moved towards half-time.
Williams' first mistake saw him concede a needless corner five minutes before the break, but Branislav Ivanovic headed over after climbing highest among a melee of players.
And then Cech was forced into action, turning a powerful Nathan Dyer shot around his left-hand post as Swansea continue to be unfazed by the challenge they faced.
Schurrle was then cautioned for a lunge on Angel Rangel in the final notable action of a scoreless opening period.
Jose Mourinho made a double substitution for the second half, sending on Oscar for Ramires and Eto'o instead of Schurrle.
Eto'o had a glorious chance to fire Chelsea ahead in the 58th minute, yet his miss from just 10 yards out epitomised the Blues' blunt attacking threat as Swansea remained on level terms.
And then a thrilling break out from a prolonged spell of defending almost produced the opening goal as Wayne Routledge's shot deflected off Chelsea skipper John Terry and bobbled wide.
But Ba finally destroyed Swansea's hopes of a point, to ensure Chelsea retain hopes of title silverware as they survived a late Swans flurry that saw Cech pressed into action.