The Swans now sit third, one point shy of an automatic promotion place, while Forest's seventh game without a win means they occupy sixth place, seven points off second.
Rodgers said: "I know what Fabio is about. He has a natural instinct to score goals but his honesty, work-rate, movement and attitude was fantastic.
"He's made a terrific start and you could see the reason why I have brought him to the club. Everyone can see his qualities and talent.
"His first goal was a sublime finish and the second one was wonderful because the ball came back to him quickly. They were finishes of top players.
"I'm really pleased for him. We were magnificent and it was a fantastic performance from the team."
On the prospect of signing Borini permanently in the summer, he added: "He has enjoyed his first couple of days here.
"I have spoken to his agent and Chelsea and if he becomes available at the end of the season then we will take a look at it. Let's wait and see what happens."