The visitors equalised early in the second half, but City missed a host of chances to have comfortably won the game.
Gary Monk made a welcome return to the starting line-up and the Swans were also boosted by the appearance of Darren Pratley on the bench.
Jordi Gomez was brought down on the edge of the area on the fourth minute but Mark Gower lifted the ball high over the bar.
Charlton responded and Jonjo Shelvey fired over after a mistake by Alan Tate.
Play switched to the other end and Guillem Bauza was set free by Dyer only to miss the target from close range.
Swansea skipper Monk lasted only 27 minutes on his return to action before being replaced by Alberto Polo Serran.
Nicky Bailey became the fourth Charlton player booked when he brought down Gomez in the 37th minute, but the Swans could make nothing of the free-kick.
But they deservedly took the lead four minutes from the break when Bauza threaded a ball beyond Graeme Murty and Dyer ran on to fire past Rob Elliot in the visitors' goal.
The second half sprung to life in the 53rd minute when Shelvey had a first-time shot pushed away by Swansea keeper Dorus de Vries.
Two minutes later Charlton were level when Bailey's 20-yard shot appeared to deflect of Tate before flying into the corner of the net.
The goal lifted the visitors and for a time they pressured the Swansea defence but without carving any openings.
It was not until 20 minutes from the end that the Swans recovered their passing game to once again make space in the Charlton half.
Serran became the seventh player to receive a yellow card in the 78th minute and the resulting free-kick from Bailey rocked the home crossbar.
Six minutes from time Dyer met a cross from Jason Scotland, but somehow ballooned the ball over from two yards out and a further chance went begging just before the final whistle, when Gomez scooped over with only the keeper to beat.