There was potential for an upset at Craven Cottage, where the hosts had won their last four games, while the Saints were winless in nine Premier League matches.
However, the visitors had the clearest chances from start to finish and, after Shane Long had a goal disallowed, Ward-Prowse struck on 29 minutes with an instinctive left-footed finish after the ball fell to him in the box.
Pierre-Emile Hojbjerg and Jack Stephens missed opportunities to secure the result in the second half, while defender Denis Odoi squandered Fulham's best chance to equalise and force a replay at St Mary's Stadium.
Relive how the action unfolded in West London with our minute-by-minute updates below.
Subs: Norman, Djalo, S.Sessegnon, De la Torre, Kebano, Ojo, Kamara
Subs: Forster, Targett, Lemina, Tadic, Redmond, Sims, Gabbiadini
Cedric Soares, Wesley Hoedt and Jeremy Pied are also on the sidelines for the visitors.
He is quoted by Sky Sports as saying: "This January I am really disappointed with the information around us. We don't have any clear targets and, for the direction of the club, I have some information that we are strong enough to fight for the important targets.
"I don't believe this. I am really disappointed with our situation and I expect the club will change their opinion. I don't know if we don't have the money or don't want to spend the money, but we are not competitive enough for the period in front of us.
"The solution? I don't know, but probably if the coach doesn't trust that we are strong enough then probably an option that the club must be thinking about is sacking the coach, and this can be a good option for the club."