The Blues took the lead in the third minute, Cesar Azpilicueta’s cross finding Abraham, who guided the ball into the bottom corner.
Norwich equalised three minutes later, Teemu Pukki squaring the ball for Todd Cantwell to tap in his first Premier League goal.
Frank Lampard's side restored their lead after 17 minutes, Mason Mount cutting in from the left and finishing emphatically past Tim Krul.
Norwich equalised again 13 minutes later, as Pukki latched on to Emiliano Buendia's through-ball to shoot past Kepa Arrizabalaga for his fifth goal in three Premier League appearances.
Chelsea made it 3-2 midway through the second half, Abraham collecting Mateo Kovacic’s pass to shoot low past Krul.
Although Ben Godfrey hit the crossbar for Norwich with a header, the west Londoners held on for Lampard's first win as a Premier League head coach.
Norwich: 31 Aug v West Ham (A)
Chelsea: 31 Aug v Sheff Utd (H)