The Senegalese substitute scooped home an 87th minute clincher to take the Blues through on the away goals rule, six days on from their 3-1 defeat in Paris.
Ba's strike compelled José Mourinho - who has still never lost a Champions League semi-final - to storm down the touchline, akin to his triumphant burst along the Old Trafford pitch 10 years ago when he was Porto manager.
Involved in two previous ties in which the side recovered from a first leg 3-1 away defeat, Chelsea summoned the spirit of their 2012 comeback against Napoli even without the talismanic Eden Hazard, who withdrew early on through injury.
André Schürrle opened the scoring in the first-half and hit the crossbar early after the pause as Chelsea searched for a second pivotal goal.
Oscar rattled the woodwork with a sweetly struck free-kick, but Chelsea could not force that second strike.
Mourinho introduced Fernando Torres and Ba in an attempt to double the lead, and it was the African who emerged the hero at Stamford Bridge.