Wins for lowly Portsmouth and Oxford United mean the threat of relegation still remains for the Addicks, who have lost three in a four-game winless run – and six consecutive league games against North End.
A bright start had raised hopes of a win that would have gone a long way to easing those relegation fears, with Rankin-Costello heading a Tyreece Campbell’s cross against the post just before he did find the back of the net.
The lead was short-lived, Dobbin’s controlled header ensuring the teams went in level.
Preston’s increasing pressure and threat led to them going ahead with 25 minutes left when Dobbin burst forward on the counter-attack and found Potts, who scored his second goal in six days – following up on his goal against QPR on Monday.
Charlton almost snatched a point late on but keeper Daniel Iversen pulled off an excellent stop to deny Lloyd Jones and earn his side just their third win in 16 Championship matches.