Fletcher started to repay his big-money fee following his move from Wolves as he struck twice in the first half, either side of Wayne Routledge's effort for Swansea.
Michu scored his fourth goal in three games for the hosts to level matters, but they then lost defender Chico Flores to a red card for a wild high challenge as the game ended all square.
But prior to all that action, Neil Taylor had already been carted off on a stretcher after his leg buckled awkwardly under him following a challenge with Craig Gardner.
The incident, which is believed to have left Taylor with a broken ankle, could barely have come at a worse time for Swans boss Michael Laudrup, just a day after the transfer window closed.
Buoyed by their excellent recent form Swansea had started with purpose, and Routledge had a penalty appeal turned away while Dyer had a shot saved by Simon Mignolet.
But they were dealt a terrible blow by Taylor's injury after 16 minutes.
The Team GB left-back required prolonged treatment and was given oxygen as he received treatment on his ankle.
The home crowd were furious that Gardner was not punished over the incident by referee Roger East, making his Premier League debut, but there appeared to be no malice from the Sunderland player.
John O'Shea did little to help matters by choosing to try to put his viewpoint across to spectators near the incident.
The atmosphere was muted following Taylor's departure although the hosts continued to press, with Dyer again being denied by Mignolet.
Swansea had benefited from defensive errors in their wins over QPR and West Ham, but they were hurt by a mistake of their own five minutes before the break.
Ashley Williams' stubbed his back pass into the ground and Fletcher ran in on goal to despatch a superb low finish beyond Michel Vorm.
Swansea levelled as six minutes of stoppage time began. Dyer and Routledge, who have started the campaign brilliantly, linked again as the former's scooped pass allowed the latter to flash home a volley.
But parity did not last to the interval as Sebastian Larsson's free-kick evaded everyone apart from Fletcher, who tapped home at the back post to give the visitors the half-time lead.
Mignolet pushed away a Dyer volley after more excellent build up play from the hosts, while Leon Britton volleyed over from a rehearsed corner routine shortly after the break.
The ill-feeling between the sides had been bubbling under the surface since Taylor's injury, but Sunderland boss Martin O'Neill was apoplectic when Flores escaped punishment for a studs-up lunge on James McClean with the home side then ignoring the stricken player by carrying on despite the Wearsiders urging them to put the ball out.
Mignolet did brilliantly to get down to a Jonathan de Guzman free-kick moments later after Gardner, already on a booking for a first-half tackle on Routledge, fouled Danny Graham.
Michu came close to adding to his tally as his header into the ground from De Guzman's cross bounced up and just over the bar.
But the same combination found the leveller on 66 minutes, De Guzman's delivery was again pinpoint and Michu's powerful header gave Mignolet no chance.
Swansea's comeback efforts were stymied when Flores then saw red for a wild high-footed challenge that grazed substitute Louis Saha's head.
Despite the numerical disadvantage Swansea dominated the final 18 minutes, but could not find the winner that would have taken them top of the Premier League.