Media reports said the Brazilian left his car in the San Siro car park and walked the short distance home having already apologised to fans and team mates for failing to cling onto the ball before Mario Gomez scored in the last minute.
"A bad night can happen," Inter president Massimo Moratti told reporters on Thursday after the repeat of last year's final.
"Julio Cesar has performed so many miracles in the past."
Moratti said having Diego Milito out injured and fellow striker Giampaolo Pazzini ineligible had cost the holders dear but that little should be read into all three Italian sides losing home first legs in the last 16.
AC Milan were beaten 1-0 by Tottenham Hotspur and AS Roma, who have since lost coach Claudio Ranieri, went down 3-2 to Shakhtar Donetsk.
"Last night was a good game, it could have finished either way," he said. "We are three different teams with particular opponents, it can happen. When we lose there's a crisis and when we win, we are the best in the world. We need a bit of balance."
Inter centre back Andrea Ranocchia went off with a knee problem but club doctors said they hope he can be fit for the return leg in Germany on March 15.
