The 20-year-old agreed to a €64million (£58m) move from Borussia Dortmund in January before finishing the season on loan at the Bundesliga club.
Pulisic has now joined his new Chelsea team-mates for pre-season and made his first appearance for the club in a 1-0 defeat to Kawasaki Frontale on Friday.
Hazard left Stamford Bridge when he signed a five-year deal with LaLiga giants Real Madrid last month, and despite playing in a similar position to the Belgium international, Pulisic is adamant that he will not be weighed down by the inevitable comparisons.
"I'm not here to compare or anything," he told Sky Sports News.
"Obviously Eden was amazing and he did so well for this club, as you could see last season and the whole time he's been here. He was an incredible player.
"For me, it's coming in as my own player, proving myself and doing the best I can to help this team.
"Right now, it's going in and helping the team whatever way I can. It's just bringing an intensity into the training first of all and earning respect in the beginning.
"Then I want to make an impact as much as I can on the pitch."
Pulisic will hope to continue his preparations for the 2019-20 campaign when the Blues take on Barcelona in a friendly on Tuesday.