In a frantic start, Abraham opened the scoring inside three minutes for the Blues, whose starting line-up had an average age of 24 years and 208 days - their youngest in the Premier League since February 1994.
But Norwich equalised just 156 seconds later, Todd Cantwell finishing from close range from a Teemu Pukki cross.
Mason Mount restored the Blues' lead midway through the first half with his second goal of the season, but the Canaries soon pulled level once more as Pukki slotted in from a tight angle for his fifth of the campaign.
Chelsea dominated the start of the second half and were rewarded through Abraham's second - an unstoppable drive past Tim Krul - before Kurt Zouma had a goal ruled out by the video assistant referee after substitute Olivier Giroud was judged to have fouled the Norwich keeper.
Abraham said at the final whistle: "We knew coming here was never going to be easy, it's always tough. We started off on the front foot, we knew we had to. We conceded but we came in to dressing room and knew there would be more chances.
"It is something I have always dreamed about doing. To score for my childhood club. To get into the first team and to start scoring goals. I am just happy I can help my team."
On getting three points, he added: "I wouldn't say it's a relief but we knew we had to win. The first two games of the season were tough. We came here knowing we needed the three points."