Inter lost 3-2 in Milan to lose the tie 5-3 on aggregate, the latest blow in a campaign which could also see the Nerazzuri miss out on a European place as they sit seventh, nine points adrift of the top three, going into the final weeks of the campaign.
Despite the public support of club president Massimo Moratti this week, the pressure on Stramaccioni appears intense with banners from the club's ultra fans on the Curva Nord last night supporting former coach Jose Mourinho.
But after the game, he was defiant. "I do not give a damn about rumours about my future," he said. "As long as I'm on the Inter bench I will give 110 per cent for these colours.
"The newspapers can pull out the names of 127 different coaches, I do not care. You pull out a name, that coach renews with his team, and you pull out another. Let's just help renew the contracts of every other coach."
Inter took the lead on the night through a superb Jonathan goal, but former Inter trainee Mattia Destro scored twice in 14 second-half minutes to put the tie firmly in Roma's hands.
Vassilis Torosidis hit a wonderful third for the visitors and Ricardo Alvarez rounded out the scoring with an 80th-minute consolation for an Inter side missing several first-choice names, including Esteban Cambiasso, who joined the long queue for the treatment room when he was injured in the warm-ups.
Insisting that his team should be congratulated for their efforts in defeat, Stramaccioni said: "We could not give any more. There was a period where things went wrong for us."
Moratti also spoke after the game and echoed the thoughts of his coach. While appearing to suggest Stramaccioni would at least see out the season, he stopped short of making any further guarantees.
"I congratulated the team, who gave all they could at this time," he said. "There will be reflections about the future, but for now we have to play these final six games.
"We cannot think that was has happened has been an accident, but we must consider the reasons that have caused it."
Asked about the Mourinho banners, he added: "The Curva is free to express itself as it wishes, but the company is also free to make its own choices."