Leandro Trossard says Albion were good value for their sixth win of the season and fifth away from home.
The Belgium international put the finishing touch to a lovely move to see off the Saints after Lewis Dunk's header was cancelled out by Che Adams' volley in the first half.
The result moves Albion up into 16th – three points clear of Fulham who occupy the final relegation spot.
“They're three very important points,” Trossard said. “We finally got what we deserved which hasn't always been the case this season.
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“We were the better team and to get the win was a relief but now we want to build on it.”
Trossard's third goal of the season was also his second match-winning contribution after he got the only goal in the win over Tottenham at the end of January.
“It's always nice to score, but especially in games like that when we needed a result. We always create chances, but we haven't always converted them this season so we're really happy to put that right today.”
“Adam is an important player for us with his experience but he also has great quality on the ball and as a team we need him. It's nice to get the confidence from that winning feeling. We will enjoy today and look to build on it in our game next Saturday against Newcastle.”