Abraham's brace fired Frank Lampard to his first Premier League victory as Chelsea manager and erased some of the doubts about his misfiring strikers.
None of Chelsea's forwards had scored in the Premier League this term until Saturday when Abraham bagged his first goals for the club in the 3-2 win.
It was a special afternoon for Abraham, who is one of Chelsea's prized homegrown assets.
Loan spells at Bristol City, Swansea and Aston Villa helped Abraham hone his craft, but when he missed his penalty in the European Super Cup shoot-out defeat against Liverpool, the 21-year-old was racially abused on social media.
It was a nasty experience for Abraham, who is aware of the doubters and is determined to silence them.
"I have always believed in myself," Abraham told Chelsea's official website.
"I know I have had a bit of a sticky time at the start of the season, but I'm someone who doesn't let that affect me. It drives me on.
"Now I have scored two good goals and hopefully I can keep scoring for Chelsea.
"This is football at the end of the day. You just have to enjoy it as much as you can. It's not a long career, so while you're out there you just have to give your best and just enjoy it."
Abraham opened the scoring with a fine finish on three minutes, with fellow youngster Mason Mount also hitting the back of the net in an entertaining first half which saw Norwich draw level twice through Todd Cantwell and Teemu Pukki.
After scoring his first goal Abraham, who settled the contest with a well-taken second-half strike, celebrated by embracing Lampard and paid tribute to his new boss.
"It's always nice to have that support behind you, especially how he supports the young lads," he added.
"He gives us confidence. When a coach believes in you, you just want to do your best and give 100 per cent."