PSG were heading for a first appearance in the last four since 1995 when Ba bundled in a loose ball after 87 minutes to clinch the tie, which ended 3-3 on aggregate.
Chelsea went ahead in a tense end-to-end battle at Stamford Bridge with a superbly-taken goal from substitute Andre Schuerrle after 32 minutes and they also hit the bar twice in two minutes early in the second half.
PSG, who won the first leg 3-1, dominated early on and Edinson Cavani missed two good chances in the closing stages which had the supporters on the edge of their seats.
Chelsea, despite losing playmaker Eden Hazard to injury after 17 minutes when Schuerrle came on, finished strongly and Senegal striker Ba’s late finish made them were worthy winners on another memorable European night in west London.
“It all happened so quick,” Ba told ITV. “I saw the ball and it was in the goal. I just do what I have to do when I get chances. I didn’t have chances this season but tonight I took it.”
Chelsea manager Jose Mourinho added: “Demba made a crucial finish for us and I think very much deserved. The team that decided to defend was punished and the team that played with their heart desered to go to the semis.”
Schuerrle scored just as Chelsea were establishing themselves following an opening half-an-hour in which PSG looked the more organised, composed and quicker side.
The visitors had taken an early grip on the midfield where Thiago Motta, Blaise Matuidi and the outstanding Marco Verratti controlled possession while forwards Lucas and Edinson Cavani kept pressure on the Chelsea backline.
But despite looking the better side, PSG did not trouble Chelsea keeper Petr Cech and the first real chance came at the other end where visiting keeper Salvatore Sirigu needed a late leaping dive to keep out a deflected Frank Lampard free-kick.
Four minutes after that Chelsea did score when Germany international Schuerrle, left unmarked in front of goal, executed a great delivery to put the hosts ahead.
Chelsea visibly gained in confidence after the goal and they started the second half stronger too, twice coming close to extending their lead on the night within 10 minutes of the break with two efforts that struck the bar.
The first came from Schuerrle whose powerful curler left Sirigu stranded but bounced back into play. Within a minute PSG conceded a freekick just outside the box with Oscar cracking the ball against the bar with his effort.
Cavani had two good chances to at least gain a draw for his side but Ba did get on the scoresheet which was enough to put Chelsea through and send Mourinho racing down the touchline to where his players were celebrating in the corner.
“This competition means a lot to us,” said Chelsea defender and captain John Terry, who was suspended for the final when Chelsea won the competition for the only time in 2012.
“The experiences we have had keeps you fighting and believing. Winning it was the best feeling ever. These big performances are what we live for.
“Everyone doubted us tonight but we showed great character and fight to come back. It all paid off,” he added on ITV ahead of Friday’s semi-final draw.
(Reporting by Mike Collett; Editing by Ken Ferris)