The pressure has been turned up on league leaders City, with their 1-0 defeat at Everton on Tuesday night coupled with Manchester United's 2-0 victory over Stoke meaning that Roberto Mancini's side now top the table by goal difference alone.
It remains to be seen whether they can hold off United's challenge, but Kompany is confident City's players will not buckle, and said: "If there is one thing I know about our team it is that we are mentally strong."
He added: "We have been living with this pressure for the past three and a half years [since the club's Abu Dhabi takeover in 2008], and it has not affected us. If anything it will give us a boost to carry on and if people want to underestimate us, then please do so."
The game at Goodison Park marked Kompany's return to the City team after a four-game suspension following his red card in the 3-2 FA Cup third-round defeat to United last month.
In the defender's absence, the team managed to win twice in the league but were beaten over two legs by Liverpool in the Carling Cup semi-final.
With the Everton loss included, it has been a frustrating period for Kompany, but the Belgian is feeling optimistic about what lies ahead and - with the visit of Fulham to the Etihad Stadium coming up in the league on Saturday - eager to push on.
"It's not been enjoyable but I have been looking at the team and still feel that we have done a good job," Kompany said. "Now we have to look forward, and looking forward I see good things for us, and that is what is important.
"I have always said about our team that we always bounce back very well, and if for some reason you are not lucky and lose a game - giving due credit to Everton - you just have to carry on. On another day we would have won the game.
"My role now is to say to the guys that this one is over, and it's on to the next game, and a clean sheet. We start from the beginning again."