Pulisic was signed in January from Borussia Dortmund but returned to the Bundesliga club on loan, meaning he is Chelsea's only new recruit for the coming season due to their transfer ban.
With star man Hazard having departed for Real Madrid, there is pressure on the United States international to perform.
But Pulisic has insisted previously he does not want to be compared to Hazard and says he would prefer to forge his own reputation – something he believes has already started.
Pulisic scored twice in a pre-season friendly win over Salzburg in what Chelsea fans hope will be a sign of things to come.
"It was a big relief to score my first goals for Chelsea," he said.
"Being a new signing and coming into the club, I felt I needed to prove myself and wanted to show the guys early on, especially my team-mates, that I can play too.
"Hopefully I have done that a bit and earned their respect.
"[Scoring] is what I'm out there to do. I'm in an attacking position. I want to score goals, I want to help them with assists and do a lot of good things. My work defensively is important too.
"I'm not trying to fill the gap left by Eden, he was an unbelievable player. I want to be my own player, to look at me like that, do the best I can and perform the best I can for the team.
"It's pre-season now and I still have an opportunity to prove myself when the real season comes around."
Frank Lampard's side kick off the new Premier League season at Manchester United on 12 August.