The American's transfer from Borussia Dortmund was finalised in January, but he finished the season in Germany - meaning he had to wait eight months to finally make his debut.
"Just to be in England, to be here in London, playing for Chelsea, it's awesome. I have a confession to make about that, actually," he said in Saturday's matchday programme for defeat to West Ham.
"During the summer, I had a short time off after playing for the national team in the Gold Cup, and I had this Chelsea jersey at home that was made for me when I signed back in January.
"I was wearing it around my house all the time - I just wanted to be here so bad, I just wanted to get started because I was excited for the challenge, excited for everything about it."
On comparisons with former Blues star Eden Hazard, now with Real Madrid, Pulisic added: "I have myself to worry about and I keep high enough standards for myself, so the pressures from outside don't really affect me as much.
"I expect a lot of myself and I have good people around me who care - the coaches, my team-mates - that are more important, so I try to focus on that."