The Blues put themselves under pressure by not extending their two-goal lead against Lille on Tuesday, eventually holding onto a 2-1 victory.
"I'm really happy to just be a part of this team,' Pulisic said after the game.
"When we were flowing like you could see in the first half it was really nice but in the end, if we're being critical on ourselves, we need to be even more clinical, myself as well.
"After they scored we made it closer than it needed to be. We should have scored more goals and made it much easier on ourselves but we got the result so it was okay and in the end, we're really happy to go through."
The win saw Chelsea advance to the knockout stages, which certainly pleased the 21-year-old winger.
"It's an amazing thing to go straight to the knockout stages," he added.
"We're a confident bunch of guys and we feel we can go up against anyone so we're really excited for the next round because the knockout stages of the Champions League is where you want to play."