Huddersfield and - sadly - West Ham and Palace have points on the board. They have six more of them than Southampton. Southampton may have a game in hand on Palace, but games in hand aren't points unless you win them. Plus Palace have Stoke and West Brom in their last two games, both of which will be down by then.
As things stand, for Southampton to overtake Huddersfield/Palace/West Ham, they need to score six more points in four games (plus goals in the case of Palace). If the three stick to approximate form, they'll score roughly four points each, so Southampton need to score ten - that's three wins and a draw.
Huddersfield do have a bit of a mince run-in, but they play Everton (who are rubbish) and Chelsea and Arsenal who may well be on the beach by then with no way to catch up to the team above or be caught by the team below. Manchester City will hurt them, but they'll have already met their 100 goal target so Pep may experiment with some younger players to give the first team some additional summer rest before the World Cup. Or he may want to go for the 100pt target. Four points is not impossible there. Palace's run in is easy. West Ham also play Manchester City, but before Huddersfield do, and may face the full force still. And they have Manchester United, who need to win to keep Liverpool at bay.
That leaves Southampton. They also play Manchester City, and if they lose to them, they can't score the 10pt they likely need to pass the other teams - but it's their last game of the season and City might be on the beach as above...
However, they can still catch Swansea City, and Swansea play Southampton on the second-last day of the season. Between now and that fixture, Southampton play Bournemouth and Everton, while Swansea play Bournemouth and Chelsea. Chelsea will still be needing the points to secure fifth and won't have given up hope of catching Spurs in fourth yet - and as the FA Cup final is two weeks later, may not rest players.
That may put Swansea in deep, Solent bobbar, saved only by the fact Southampton couldn't score in a whorehouse and Swansea's own last game of the season is Stoke...