The hosts went ahead in the first half through Andre Schurrle and hit the woodwork repeatedly after the half, with the Blues finally rewarded for their efforts with a late winner.
Jose Mourinho's men came into Tuesday's home leg of the quarterfinal tie trailing 3-1, and an early injury to Eden Hazard looked set to make its task even tougher.
However, the Belgian's replacement - Andre Schurrle - gave the home fans hope by finding the net 14 minutes after coming off the bench.
PSG looked a shadow of the side that cruised to victory against an off-color Chelsea at the Parc des Princes last week, and was walking a tightrope as Schurrle and Oscar both found the woodwork in quick succession after the break.
However, Ba - a 66th-minute replacement for Frank Lampard - sent Stamford Bridge into raptures and Jose Mourinho sprinting down the touchline with a winner three minutes from time, as the Londoners progressed on away goals.
The result means Mourinho has now been successful in each of his nine European quarterfinal ties.
PSG was forced to make do without talisman Zlatan Ibrahimovic, who limped out of the first leg with a hamstring problem, while Chelsea fans will have been delighted to see a striker at all - Samuel Eto'o back in the starting XI after Mourinho's decision to start without a recognized frontman backfired in Paris.
Mourinho probably should have been the one sweating over his side's future in the competition, but it was opposite number Laurent Blanc who emerged for the first half chewing on a toothpick in an apparent bid to calm any pre-match nerves.
His anxiety appeared misplaced, however, as the visitors made an accomplished start to the clash, preventing their hosts from taking control.
And Chelsea's prospects appeared to suffer a huge blow in the 18th minute when reported PSG transfer target Hazard limped from the field to be replaced by Schurrle.
Lampard was first to really test Salvatore Sirigu in the PSG goal after 28 minutes - his free kick deflecting off the wall and forcing the Italian into a one-handed diving save.
But Chelsea was ahead four minutes later, when David Luiz nodded a long throw onto the boot of substitute Schurrle, who made no mistake in slotting past Sirigu from the center of the penalty area.
The German - ineffective as Chelsea's most advanced player in the first leg - went down in the box under the challenge of Marco Verratti during the hosts' next attack, but the referee ignored home claims for a penalty.
The men from the French capital posed little threat after going behind, and could have been two down when Schurrle rifled an effort against the crossbar, before an Oscar free kick suffered the same fate soon after.
It did not appear to be Chelsea's day, but, after Edinson Cavani had wasted a golden opportunity to put the result beyond doubt, Ba became an unlikely hero.
The Senegal striker, who has often been out of favor this season, got the final touch on Cesar Azpilicueta's deflected delivery to divert the ball past Sirigu to spark joyous scenes at the Bridge.