Substitute Brandao scored in stoppage-time to send the French side sneaking through on away goals after a 2-1 loss at Inter earned a 2-2 aggregate draw.
Marseille had also snatched the first leg thanks to a stoppage-time header from Andre Ayew.
On Tuesday, Diego Milito's scrappy goal 15 minutes from time had given Inter hope after the hosts had huffed and puffed before finally breaching Marseille's defence.
Brandao, who spent most of last year on loan in his native Brazil before returning in January because of an injury crisis, then exposed Inter's defence when he turned on a long ball and confidently netted.
"We had difficult moments, we couldn't score but when I put him on Brandao said to me: 'I am going to score, I am sure,'" Deschamps told reporters.
"To be in the Champions League quarters is fabulous. It will be a bit mad in Marseille," added the former Juventus coach, whose side meet Olympique Lyon in the French League Cup final next month and are also in the French Cup quarter-finals.
"The team could not have given any more even if we could have had a little more luck. It is always the coach's fault, however, we are paid to take the blame," Ranieri said.
"We have been condemned and I think unjustly, we created more chances in the 180 minutes but this is football, the ugliness and beauty of football."
The hosts had the first glut of chances at an apprehensive San Siro when Wesley Sneijder's shot from close range was somehow saved by Mandanda, who soon after brilliantly blocked Diego Milito's chested effort from point-blank range.
Marseille, playing in red, gradually grew into the game and a Loic Remy effort flew just wide before Jeremy Morel fired past the post and Souleymane Diawara's header almost crept in.
