With the Blues only sitting in ninth place in the Premier League standings, Pochettino has not endured the ideal start to his reign at Stamford Bridge.
However, the West Londoners have booked their place in the final of the EFL Cup after overturning a 1-0 deficit against Middlesbrough by winning 6-1 in the reverse fixture.
After fans witnessed Chelsea fail to trouble the scoresheet at the Riverside Stadium earlier this month, frustration threatened to boil over within the travelling support.
In sharp contrast on Tuesday, Chelsea scored four goals in the first half to effectively end the tie as a contest, before two more strikes followed during the closing stages.
As well as an own goal from Jonny Howson, there were goals from Enzo Fernandez, Axel Disasi, Noni Madueke and a double from Cole Palmer, with Morgan Rogers netting a consolation for the visitors.
Ahead of a showpiece against either Fulham or Liverpool at Wembley on February 25, Pochettino has called on the club's fanbase to find a balance of remaining patient in search of trophies but believing in what his players are trying to carry out on the pitch.
Speaking to Sky Sports, Pochettino said: "Sometimes we can think about the history of Chelsea and the missing players. We are young and we need to grow.
"These types of games are a good experience for young players and it is about now growing and it will help us to compete. We are building a very good team. The mentality is good. We have plenty but we need time.
"I think the performance, I was never disappointed with the performance - it was because we were not clinical. Today I think is our ninth game unbeaten and it is really good. It is a new project. We are building a team and sometimes it is normal to play worse. We played well and we didn't win but now it will be a big boost for us.
"We don't have much time to prepare for Aston Villa and also with many injuries, I think we are very happy. People need to believe and trust in us and we go to the final with the chance of winning the first trophy of the season."
Chelsea are back in action as soon as Friday when they play host to Aston Villa in the FA Cup fourth round.