Chico Flores was sent off for a second bookable offence after just 16 minutes, leaving the relegation-threatened Swans with a mountain to climb if they were to get anything from the game.
It seemed like they could get at least a point, however, with Chelsea proving wasteful in front of goal until Ba's 68th-minute strike broke the deadlock. From there, the Blues closed the game out and came away deserved winners.
With a precious three points in the bag, the result leaves Jose Mourinho's side breathing down the neck of Liverpool at the top of the table and putting them firmly back in the title race.
And with Chelsea traveling to Anfield in two weeks, this victory has whet the appetite for what will surely be a game that decides where the Premier League trophy resides come May.
Here are six things we learned from Chelsea's trip to the Liberty Stadium.