A goal from Salomon Kalou after 41 minutes was enough to seal a narrow victory against struggling Bolton at the Reebok Stadium.
Following Jose Mourinho was always going to be a tough task but Grant said: "I do not want to live in football without pressure.
"When you are at a big club you always have pressure if you want to achieve the best.
There is extra pressure if you want to change the way Chelsea play.
"It is a big responsibility but I like it. I like to put pressure on myself. The name of the game is pressure. It is not a game for weakness."
Grant added: "The target long-term is to play good football. We will see how we develop from month to month.
"The base before was good but that has not been the case this season. I want all the forwards to play with freedom but also responsibility.
"Sometimes, however, you need the points, especially when you play away. The target for today was to take three points and we did that which was important.
"We deserved to win as we created more chances. It was not easy. Bolton play direct and have physical players."
Grant said: "For Frank to play for 90 minutes after six weeks out is good for him and good for us."