The midfielder scored twice in extra time as Chelsea came from 2-0 down to beat Leicester 4-2 in the EFL Cup third round on Tuesday
He is yet to start a Premier League game this season, with his only two starts coming in the cup against Bristol Rovers and the Foxes
Fabregas scored in the first four minutes of extra time after Gary Cahill and Cesar Azpilicueta cancelled out Shinji Okazaki's first-half double before Leicester's Marcin Wasilewski was sent off in the 89th minute.
Chelsea go to Fabregas' former club Arsenal in the league on Saturday and the Spain World Cup winner wants to show Conte, who arrived in the summer, why he is wrong to leave him out.
"It's a new situation for me in my career after 13 years of basically playing all the time for my club and national team," he said
"And it's been tough - I'm not going to deny it
But the last thing I will do is complain or put my hands up and give in.
"Something from inside is coming out stronger than ever and I just feel like a little boy in training trying to win my spot back as I used to do at Arsenal when I was 16
Hopefully I will achieve it again.
"My coach deserves respect because I know he wants to win and if he doesn't play me there is always a reason
"Hopefully I can open his eyes and make him see that I deserve to play."
Fabregas spent eight years at Arsenal before a 2011 move to Barcelona, who sold him to Chelsea two years ago, and admits he is desperate to play against his former club
"No matter what I hear, what they say to me, Arsenal will always be a fantastic special place in my heart, " he told the Evening Standard
"It's too early to know whether I will start or not
"Hopefully I will
I'm not going to deny that I want to play every single minute."
Fabregas' quickfire double ended Leicester's EFL Cup hopes after Okazaki had put them 2-0 ahead inside the opening 35 minutes with his first goals since March
The striker told the club's official site: "For me, scoring two goals is important because I'm a striker
I didn't score for a long time and I scored, so it's okay
It's very important for me to keep going."
Source : PA
Source: PA