City recorded their first victory in four games, while their promotion rivals are now winless in seven attempts.
The on-loan Chelsea striker, who arrived earlier this week on a deal until the end of the season, coolly opened his Swans account from close range shortly after the impressive Scott Sinclair had registered his 17th league goal of the season with a sublime virtuoso effort in the 21st minute.
City were firmly in command and Italy Under-21 international Borini smashed against a post, before Kris Boyd - making his first Forest start - gave the visitors hope with a neat finish on the stroke of half-time.
But it did not last long as man-of-the-match Borini capped a debut to remember with a simple second just before the hour mark.
Paul Anderson added an unjust slant on the scoreline in injury time, before then heading against the woodwork as Forest came close to snatching a undeserved point.
Victory was no less than Swansea deserved from a scintillating display, with a large gulf in class between them and out-of-sorts Forest at the Liberty Stadium.
Brendan Rodgers' side started on the front foot amid a fast-paced opening to proceedings, with Reds defender Paul Konchesky appearing fortunate to get away with a challenge on City winger Nathan Dyer in the penalty area.
The first real chance of the afternoon came in the 17th minute as Sinclair saw his effort turned around a post by Lee Camp.
Boyd forced a solid save from Dorus de Vries at the other end as the visitors momentarily got some kind of foothold in the encounter.
But it was then that the hosts broke the deadlock courtesy of a great solo goal from Sinclair.
The ball broke kindly for the former Chelsea winger around 15 yards into the Forest half on the left, and he ran unchallenged all the way across to the right-hand side of the penalty area before firing back across Camp and into the net.
Five minutes later and it was 2-0 as Borini bagged his first goal in the white of Swansea.
Angel Rangel got to the byline and cut the ball back for the 19-year-old Italian who saw his initial effort blocked, but the rebound bounced back into his path and he kept his composure amid four or five lunging challenges to pick his spot.
Swansea were firmly in control and Borini drilled an effort against the woodwork seven minutes before the break.
But, out of the blue, Boyd gave Forest hope just before half-time as he received a pass from Nathan Tyson on the byline, turned and fired past De Vries who could only get a hand on it.
And the Reds could have been level before the break as Luke Chambers' header came agonisingly close.
Stephen Dobbie forced Camp into an early save at the start of the second half before Mark Gower drilled across goal as Swans continued to dominate after the break.
Sinclair was constantly causing problems when running at the Forest defence and he created City's third with another dazzling charge which resulted in Camp parrying his effort into the path of Borini, who was left with his second simple finish.
Borini could only find the side-netting in the 80th minute with his hat-trick begging.
But Swans were made to sweat as Paul Anderson grabbed an injury-time second for Forest, before heading against a post.