Trailing 3-1 from last week’s quarter-final first leg in Paris, Chelsea trimmed PSG’s aggregate advantage through a 32nd-minute goal by Andre Schuerrle, but the French champions remained on course for the last four.
The hosts hit the bar through Schuerrle and Oscar early in the second half, but with time running out, substitute Ba bundled home from close range to take Chelsea into the semi-finals for the seventh time in 11 years.
Chelsea progressed on the away goals rule after a 3-3 aggregate draw, two years after pulling off a similar feat by coming back from a 3-1 loss to beat Napoli 4-1 in the last 16 en route to success in the 2012 competition.
Ba’s goal preserved Chelsea coach Jose Mourinho’s unbeaten record in European quarter-finals and his glee was evident as he sprinted down the touchline in a manner reminiscent of his celebration when Porto eliminated Manchester United in 2004.
PSG had been on the verge of the semi-finals for the first time since 1995 and the only consolation for Laurent Blanc’s heartbroken side was that they remain on the brink of a successful Ligue 1 title defence.
Chelsea’s need for goals prompted Mourinho to hand a start to Samuel Eto’o, who had missed the previous three games with a hamstring injury, and he saw a snapshot blocked in the ninth minute.
Branislav Ivanovic headed wide from the subsequent corner, but the hosts received an early setback in the 18th minute when Eden Hazard had to come off with an injury.
Exacerbating the Chelsea fans’ unease was PSG’s assurance in possession, with the visitors popping passes around in midfield and threatening on the flanks through Lucas Moura and Ezequiel Lavezzi.
Chelsea began to apply more concerted pressure, however, and after PSG goalkeeper Salvatore Sirigu produced a brilliant flying save to parry a deflected Frank Lampard free-kick, they broke through.
David Luiz used his back to flick on Ivanovic’s long throw-in from the right and Schuerrle stole in to steer a volley past Sirigu from 10 yards.
– PSG spurn chances –
With Stamford Bridge rocking, the hosts flooded forward. Schuerrle saw a penalty appeal waved away after flopping over a challenge from Marco Verratti, while Gary Cahill miscued a volley right in front of goal.
Chelsea resumed their onslaught after the interval and hit the bar twice in the space of two minutes, first through Schuerrle, from Willian’s cut-back, and then via an arcing free-kick from Oscar.
Blanc sought to shore things up by sending on Yohan Cabaye for Verratti and briefly PSG rallied, with Petr Cech touching a free-kick from Lavezzi around the post and Cavani heading wide from a corner.
Mourinho responded by introducing Ba in place of Lampard and the Senegalese striker soon made his presence felt, finding Schuerrle with a flick-on, only for Sirigu to field the German’s shot.
With Chelsea pressing forward, they left themselves open to the counter-attack and PSG spurned a host of chances to make the game safe.
Cavani twice shot over from promising positions, while Maxwell skidded a low shot wide and Cech had to save a drive from Lucas.
Those misses would have proved immaterial, but with three minutes to play, Cesar Azpilicueta drove a shot across the box and Ba outmuscled Maxwell to touch home and send Mourinho scampering down the pitch in delight.
Cech preserved his side’s place in the last four with an injury-time save from Marquinhos and not even a booking for Ivanovic that rules him out of the semi-final first leg could dampen the mood at a jubilant Stamford Bridge.