The Giallorossi beat the Nerazzurri at San Siro for the first time in five years - and they came from behind to do so, as Paulo Dybala and Chris Smalling got the vital goals.
And Mourinho - who watched the game from the team bus as he served a one-match touchline ban - was left delighted with his charges.
"Yes, we've spoken to the boss, he was very happy," Cristante revealed afterwards.
"It is true that in the last few years we've struggled a bit in games against other big sides, especially away from home. Then today we went behind against Inter, but we did a really good job of bouncing back and getting the three points.
"We deserved to win. I don't think it was lucky; apart from their opening goal, I think we put in a really good performance."
Alongside Nemanja Matic, Cristante helped marshall the midfield for Roma throughout.
"Matic is a top player, he played in the Premier League for so long," the Italy international said.
"The more we play together, the more our understanding gets better and more natural. Every match helps us get to know the other better and play with more fluidity."
He added: "We need to make sure we stay focused and don't create too many expectations. We can't worry about where we might end up. We know that we are a good side but we need to take things game by game.
"If we play like we did today, then we can do well. But we have to make sure we keep a level head."
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