Fletcher - a £12 million signing from Wolves - scored with two fine finishes on 41 minutes and deep into injury time in the first half, but goals from Wayne Routledge (45+1) and Michu (66) - who increased his tally to four in three games to top the goalscoring standings - were enough to maintain Michael Laudrup's side's unbeaten start to the league season.
Fletcher is the first debutant to score twice on his Sunderland debut. The only player in Premier League history to score a hat trick on their Premier League debut for a club was Fabrizio Ravanelli for Middlesbrough against Liverpool in 1996.
In truth, Swansea did enough to merit all three points from the afternoon and continued to look the likelier side even when Chico Flores was red carded for aiming a late kick at visiting substitute Louis Saha - who had replaced Fletcher moments earlier - off the ball with 19 minutes left.
Apart from the blow of ending the game a man down, Swansea also lost left-back Neil Taylor early in the match after he was stretchered off following a collision with Craig Gardner.
Taylor was taken to hospital with reports indicating that he had suffered a fractured ankle.
Swansea named an unchanged side from the team that beat West Ham United 3-0 in the Premier League last week. New signing Ki Sung-Yueng was on the bench after the former Celtic midfielder made his debut in the win over Barnsley in the League Cup second round. On-loan Kaiserslautern striker Itay Shechter was also on the bench, but £6 million club record signing Pablo Hernandez from Valencia wasn't registered in time to be included for the match.
Adam Johnson and Fletcher were both handed Premier League debuts by Sunderland manager Martin O'Neill. The pair, who signed for the Black Cats a week ago, started Tuesday night's Capital One Cup second round victory over Morecambe and retained their places for the game.
Fletcher struck against the run of play with Sunderland's first shot of the afternoon when he bounded onto a loose attempted passback by Ashley Williams before slipping the ball beyond goalkeeper Michel Vorm in the home goal.
With six minutes added to the half due to Taylor's injury and the departure of Sunderland skipper Lee Cattermole with a dead leg, Swansea were deservedly level before half-time when Routledge thumped Nathan Dyer's pass beyond Simon Mignolet.
Sunderland somehow headed for their half-time refreshment holding a 2-1 lead as Fletcher ghosted in at the back post to prod a Sebastian Larsson free-kick over the line to delight the large Sunderland travelling support.
The home side were dominant for large swathes of the second period and were rewarded when Michu evaded the clutches of John O'Shea to send a header spinning beyond Mignolet.
Flores was probably right to be red carded for a silly challenge on Saha, but the home side continued to dominate possession despite their one-man disadvantage. They almost won it in the closing moments, but Dyer shanked a loose ball wide when he seemed likely to at least hit the target.