The Blues striker was named in Frank Lampard’s starting XI for the trip to Carrow Road despite failing to score in his previous three appearances.
Abraham netted his first Chelsea FC goal in the third minute when the 21-year-old produced a great finish from Cesar Azpilicueta’s cross.
Mason Mount restored Chelsea FC’s lead after Todd Cantwell had levelled before Teemu Pukki netted his fifth goal in three Premier League games for the hosts.
Lampard stuck with Abraham in the second half and the England Under-21 international produced a clinical finish past Norwich goalkeeper Tim Krul in the 68th minute.
Former Chelsea FC striker Sutton was full of praise for Abraham after his clinical second goal against the promoted side.
“It’s terrific from Tammy Abraham,” Sutton told BT Sport.
“It’s a wonderful first touch. It’s a good run in behind. That’s what he was in the team for. That little shift across the defender.
“That’s a finish of a confident striker, it really is. It’s a wonderful finish from Abraham.”
Asked if Abraham has answered questions looming over the Englishman, Sutton replied: “He absolutely has answered questions.
“He was brought in ahead of Olivier Giroud. He started [the game] for a run like that.”
Chelsea FC loaned out Abraham to Swansea City in the Premier League last season.
The 21-year-old netted five goals in 31 games in the Premier League during his stint at Swansea.
The Blues forward hit the woodwork against Manchester United in their 4-0 loss on the opening day before he missed the decisive penalty in their shootout defeat by Liverpool FC in the Uefa Super Cup earlier this month.
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