The last time Southampton reached the FA Cup quarter-finals was in the 2004-2005 campaign, losing each of their last four fifth-round ties.
Wesley Hoedt broke the deadlock in the 11th minute as he swept the ball past Ben Foster from close range after he was left unmarked from a James Ward-Prowse corner.
In the 56th minute, Dusan Tadic made it 2-0 from a counter on 55 minutes, chipping the on-rushing Foster.
West Brom pulled a goal back through a superb left-foot volley by Salomon Rondon from the edge of the area off Grzegorz Krychowiak’s lofted through-ball.
The other FA Cup tie between Brighton and Hove Albion and Coventry, which was delayed by 15 minutes after trains were held up because of a tragic car crash at a level crossing on Saturday morning, ended 3-1 in favour of Brighton.
Brighton’s goals were scored by Jurgen Locadia (15th minute) Connor Godson (34th minute), Leonardo Ulloa (61st minute).
In Saturday’s FA Cup early kick-off, Swansea forced Championship club Sheffield Wednesday to a replay after holding them to a 0-0 away draw.