"I am not unbeaten at the San Siro any more but we qualified," he told Marseille's website after his team reached the quarter-finals on away goals when a 2-1 second-leg defeat by Inter left the aggregate scores level 2-2.
"I am happy. I do this job to live these moments," added former France midfielder Deschamps who spent five years as a player with Juventus and also coached the Turin team in 2006 and 2007.
Marseille, the only French club remaining in European competition this season, reached the Champions League last eight for the first time since 1993 thanks to a late goal from substitute Brandao.
The striker has been criticised in some quarters since returning to the club in January following a loan spell with Cruzeiro in his native Brazil.
"There have been hard times since I came back (but) tonight I thank all the fans who were very supportive," said Brandao.
"I don't know how I got the ball for my goal, then I just shot and got a bit lucky."
The only sour note for Marseille was the late sending-off of goalkeeper and captain Steve Mandanda which means he will be banned for the quarter-final first leg.
Brandao's effort cancelled out Diego Milito's opener for Inter who grabbed a second goal six minutes into stoppage time with a penalty from Giampaolo Pazzini following a foul in the box by Mandanda.
"The penalty cost us a lot because Steve is going to be suspended and we saw what he did in the first half," said Deschamps, referring to a string of saves by his keeper.
Mandanda praised his replacement Gennaro Bracigliano who has also played in Europe with former club Nancy.
"Gennaro is very experienced," said the first-choice keeper. "He played a lot of Ligue 1 matches with Nancy and European games also.
"I'm not worried. I know he will do the job."