In a frantic start, Abraham opened the scoring inside three minutes 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.
