The Saints are facing several "six pointer" battles against teams around them at the bottom of the Premier League in the coming weeks.
And Stephens knows they must pick up maximum points in as many of those games if they want to stay up.
The first step is beating Aston Villa on Saturday.
Stephens told reporters: "Every game is high-pressure but when you play teams around you it's so important to win.
"Saturday's result was a massive boost for West Ham and a big blow for us.
"We did it against Watford and Norwich but needed to kick on at Newcastle and shouldn't have lost there from the position we were in.
"Saturday wasn't good enough. We didn't create or show enough intent.
"It was so disappointing. The away games are difficult enough but these home games, especially against teams around us, we have to be a lot better.
"We need to create more chances, we need to be braver, to play and get on the ball. We didn't do that on Saturday. We need to be better all over the pitch.
"Every game is high-pressure but we need to be relishing these games, we can't hide away."