Jan Bednarek’s second goal in three games proved the winner for the Saints, who tasted success for the first time in seven Premier League matches.
It is now five without a win for the Hammers who came from behind twice to level in the second half, but to no avail.
“It is a huge win because our record here in the last five years is that we haven’t taken a point here,” Hasenhuttl told Sky Sports.
“We played well today – we never stopped believing and scored three fantastic goals. We had a bit of a push back but an excellent opening 15 minutes – and we can’t play that way for 90 minutes so it’s normal but we played well.
“We have dropped points a lot in the league this season but we showed resilience and absolutely had the right answers for everything they put on the pitch. We showed tactically a very brave performance.”
The visitors made early pressure tell in the eighth minute as Mohamed Elyounoussi fired past a sprawling Lukasz Fabianski from the edge of the area after being fed by Kyle Walker-Peters.
Elyounoussi remained a threat on the break but squandered a good chance just past the half-hour mark. After connecting with Nathan Redmond’s pass and skipping past Craig Dawson, he failed to connect properly with the finish and shot straight at Fabianski.
Fraser Forster was finally forced into action six minutes before half-time when Nikola Vlasic drove forward and struck to his left. The Saints keeper reacted well to parry, with the rebound evading everybody.
Four minutes after the break, Jarrod Bowen’s corner was won by Dawson and substitute Michail Antonio, who had returned after a bout of Covid-19, was alert to head in from close range to equalise.
Moments after Declan Rice’s shot whistled past the post at one end, Armando Broja was brought down in the area at the other. After checking the pitchside monitor following a VAR review, Kevin Friend awarded the penalty, which James Ward-Prowse converted in the 61st minute.
West Ham instantly hit back again, this time through Said Benrahma. He ghosted in at the back post to slam home Bowen’s cross three minutes later.
Yet the twists kept coming as defender Bednarek rose highest to nod Ward-Prowse’s pinpoint free-kick into the bottom corner, giving Southampton the lead for the third and final time after 70 minutes.
West Ham can’t make dominance pay in chaotic clash
David Moyes’ side were punished for a sloppy start but gradually, it was they who began to dictate the play from midway through the first half into the second.
Without Antonio, who had gone 10 games without a goal himself, they looked toothless in the first 45 minutes, while Southampton executed their counter-attacking approach superbly.
After the Jamaica international’s introduction, the Hammers were much more threatening. But they couldn’t stay level for long enough on both occasions after they equalised, constantly making life difficult for themselves on a frenetic Boxing Day afternoon.
After a bright start to the season ignited hopes of another top-four push, West Ham dropped to sixth and have their work cut out to get it back on track.
Broja, leading the Southampton attack, was a handful throughout as his side moved nine points clear of the relegation zone.