Beaten 1-0 by Queens Park Rangers in the EPL on Wednesday, Chelsea fell behind in the 22nd minute at St Mary’s when Jay Rodriguez got in behind Gary Cahill before deftly beating goalkeeper Ross Turnbull.
Interim manager Rafael Benitez handed a first start to Ba since his move from Newcastle and the Senegal international crowned his debut with a 35th-minute equaliser when he helped a Juan Mata effort over the line from close range.
Victor Moses rifled home to put Chelsea ahead on the stroke of half-time, with Branislav Ivanovic heading home in the 52nd minute before Ba completed his brace and substitute Frank Lampard netted a late penalty.
City stroll into Round Four
English champions Manchester City enjoyed a similarly straightforward afternoon at home to second-tier Watford, winning 3-0 to book their place in round four.
Carlos Tevez opened the scoring in the 25th minute via a ferocious free-kick, with Gareth Barry heading home just prior to half-time before Marcos Lopez made it 3-0 in the 90th minute.
Earlier, the FA Cup began with an upset as second-tier Brighton & Hove Albion eliminated Premier League team Newcastle United for the second year in succession.
Shikhar Dhawan birthday: Top 10 scores by an Indian on Test debut
Spotted: Shilpa Shetty, Raj Kundra at a Mumbai multiplex
Spotted: Virat Kohli, Anushka Sharma and other celebs at Mumbai airport
Photos: Salman Khan's 'special friend' Iulia Vantur at a song launch
Photos: Hrithik Roshan, Anushka Sharma, other celebs dazzle at event