Hull could have been in the lead in the 10th minute as Paul McShane whipped a cross to the back post and found David Meyler, whose header was cleared off the line by Dale Stephens.
Steve Bruce's side took the lead in the 33rd minute as Gedo scored his second goal of the week, poking home from close range after Ben Hamer came for a corner and missed the ball.
Charlton hadn't threatened David Stockdale's goal until Bradley Pritchard drove into the Hull box but he could only fire his shot just wide of the post after beating the keeper.
Minutes later, substitute Danny Haynes found himself in on goal as he latched onto a lofted ball into the area, but he couldn't beat Stockdale as he powered his shot straight into his hands.
Haynes had another great chance to get his side level as Bradley Wright-Phillips found him in the box but his effort was beaten away by Stockdale, which was enough to close out the win for the home side.