The 26-year-old was acting captain as his side faced a negative reaction to the 1-0 defeat at Stamford Bridge, with Anthony Elanga scoring the only goal.
Pochettino said his side lacked maturity after dominating possession but came away with nothing, with his team having earned only four points from the first four Premier League matches of his tenure.
Asked to give a message to supporters while on England duty, Chilwell simply said: “Patience”. He then explained: “I understand that fans expect and want us to win week in, week out — it is what they are used to and have always had — it is completely understandable.
“What the manager has come in and done is brilliant. We have all bought into what he is trying to do at the club. With such a young, exciting squad.
“I think it will be a good future. With a bit more patience, we will come good.”
Chelsea couldn’t finish their chances and made a series of mistakes which led to Elanga’s winning goal in the second half.
Pochettino was criticised for playing Chilwell on the left wing and Levi Colwill at left back, roles they aren’t typically used to.
Chilwell accepts that he needs to take increased responsibility to score the goals to make his more advanced role work. He added: “It is something new but the manager has put his trust in me to play there so I try to repay that trust.
“I am still learning as I am going. I have been playing left back and then left wing back and now I am, again, higher up the pitch. I expect the pressure on me to contribute to goals.
“That’s something I am going to work on at Chelsea over the next few games to add that goal threat. [I need to] shoot, instead of passing the ball.”
Chilwell will revert to left back under Gareth Southgate to face Ukraine in Poland on Saturday as he bids to regain his starting spot with Luke Shaw out injured.
At Chelsea, Chilwell is now considered one of the leaders in the Premier League’s youngest squad. He accepts it is his duty to help his younger teammates adapt.
Chilwell added: “It is such a young and new squad. It’s inexperienced in a way. There’s a new manager trying to impart his ideas and it is down to me and Reece to take on responsibility and be leaders in the group and guide the ship a little bit.
“Especially in the last few weeks in what could be seen as a bit of a tough time. I kind of just try to guide the ship through that, stay confident and know what talent we have in the squad. We are confident and also have a great manager.
“We are very aware [of what’s wrong] but confident we can turn it around very quickly.”