Watford's impressive attacking play threatened the Charlton defence throughout the first period but Gianfranco Zola's side finally took the lead from a set piece situation.
Tommie Hoban's run across the box gave him space and the defender's header deflected past Hamer to give the Hornets their first goal of the contest.
The visitors were dealt a cruel blow on the stroke of half-time when striker Fernando Forestieri was shown a second yellow card for diving inside the Charlton penalty area.
Despite the disappointing end to the first half, Watford regained the lead on the hour mark when Abdi curled in a superb free kick to the delight of the travelling supporters.
Watford survived a stern test from Charlton late on but Zola's men stayed firm in defence to earn an impressive away win.