The Hammers find themselves five points clear of the relegation zone after the weekend's win against fellow strugglers Saints, with Marko Arnautovic bagging an important first-half brace.
There is still plenty of work to be done in West Ham's efforts to secure their Premier League status, with 18 points still to play for between now and the end of the league campaign.
Cresswell, who played his 150th game for the Hammers on Saturday, feels that the win over Saints shows what his side are capable of and that should be the bar set for the rest of the season.
He told the club's official website: "Let's not beat around the bush, the pressure was certainly on us on Saturday, but we performed straight from the off.
"The fans were behind us, we were all as one and the first half was excellent - the best we've been all season.
"Everyone played a part. We set our stall out, got that 3-0 lead, and then the second half was pretty comfortable."