It looked like it would prove elusive until Cesar Azpilicueta's drive into the area fell for Ba to turn the ball into the net and spark delirious celebrations, led by Mourinho.
Mourinho had gambled on the fitness of big game player Samuel Eto'o, but was not banking on losing Hazard to injury after 18 minutes.
Sirigu was beaten soon afterwards as his static defence watched David Luiz flick on Branislav Ivanovic's long throw for Schurrle to sweep the ball into the bottom corner.
Chelsea maintained the intensity after the restart Nike Kyrie Red And Black and went close when Oscar's incisive pass found Willian.
Only two sides in Champions League history had overturned two goal deficits, with Chelsea involved in both contests, losing to Barcelona in 2000 and beating Napoli Nike Zoom Pegasus 32 Black
goal with a free kick moments later and PSG had chances as Chelsea surged forward looking for the decisive goal.
Mourinho believed in his side and, when Andre Schurrle swept in after 32 minutes, the Blues needed one more goal and to stretch that defensive mean streak by one more game.
Chelsea trailed 3 1 after the first leg, but were unbeaten since September at home and had kept eight successive clean sheets prior to the match.
PSG had lost just once by two clear goals in their previous 110 fixtures and boss Laurent Blanc hoped his side could impose their philosophy on Chelsea to show flair as well as grit.
Schurrle, the Belgian's replacement, was anonymous in a 'false nine' role in the first leg, but thrived in the second, hitting the bar seven minutes into the second half.
Schurrle appealed in vain for a penalty moments later when collided with Marco Verratti.
The Portuguese dressed down in a tracksuit, but meant business, hurrying a ball boy operating near his dugout.
wing is usually Hazard's domain, but much to the concern of the raucous home faithful, the Belgian was subdued and soon limped off. The man coveted by PSG shook hands with Blanc on his departure.
The Blues had set piece opportunities.
Mourinho, meanwhile, told his players to enjoy the occasion, to play with smiles on their faces, insisting the result in Paris was more due to his side's failures than PSG's prowess.
Chelsea and Jose Mourinho dare to dream of European glory once more after dramatically overturning a two goal deficit to advance to the Champions League semi finals at the expense of Paris St Germain.
Oscar struck the frame of the Nike Zoom Hyperrev 2016 Low
12 years later en route to European glory.
finals after dramatic night at Stamford Bridge
Against a PSG strikeforce vaunted even without the injured Zlatan Ibrahimovic that was no easy task, but the defence held their side of the bargain as the attackers tried to fulfil theirs.
Few Chelsea players seemed to enjoy a pulsating and disjointed opening.
A Frank Lampard free kick kicked up wickedly off the wall, forcing Sirigu to turn the ball round the post.
Oscar found some joy down the left, so, too, did Eto'o, who had a shot deflected over before the striker showed his sharpness after a three game absence with a hamstring injury by pressing Salvatore Sirigu as the goalkeeper dealt with a back pass, only to commit a foul.
Cavani and Lampard were booked in quick succession by referee Pedro Proenca the man in the middle when the Blues won the 2012 European Cup as tempers simmered.
Mourinho called on his side to repeat the comeback of two years earlier against a PSG side which included Edinson Cavani and Ezequiel Lavezzi, then of Napoli.
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