The game ended with Fulham fans protesting against manager Claudio Ranieri, with the team now 10 points from safety with 10 games to go.
Ranieri has failed to spark an upturn in results since replacing the sacked Slavisa Jokanovic in November and Fulham has now lost seven of its last eight league matches. The London club is 19th and looks increasingly likely to go back down in the Championship along with last-place Huddersfield.
Southampton, meanwhile, is among a handful of clubs aiming to avoid joining them.
With daunting fixtures against Manchester United, Tottenham and Liverpool coming up, this game had a must-win feel for the hosts, and the anxiety among home supporters was partially eased with 23 minutes played.
Wing-back Ryan Bertrand delivered a corner from the right and, after Fulham goalkeeper Sergio Rico could only punch the ball to the edge of the box, Spanish midfielder Romeu was well placed to lash home a first-time effort through a crowded penalty area for his first goal since November 2017.
Southampton doubled the lead in the 40th minute when Nathan Redmond saw his low shot turned away by Rico before the alert Ward-Prowse tapped home the rebound to claim his fourth goal of 2019.