The sighs of relief around the London Stadium at the end of West Ham’s 1-0 win over could have blown over the Orbit sculpture just outside. But for Slaven Bilic this win, which established an eight-point gap between West Ham and the bottom three, does not mean the threat of relegation has been completely dismissed.

“There’s still work to do,” said the Hammers manager after a performance that was a long way from pretty but a result that was absolutely necessary. “We opened the gap a bit more, but we opened the gap a bit more a few months ago then we lost five in a row. Four more points is the minimum we need. Forty points is normally enough.”

West Ham are probably safe now, but Bilic pointed to another side who thought they were safe as an example of how quickly things can change. “You see PSG v Barcelona,” said Bilic, referring to the Champions League tie in which Barcelona scored three times after the 88th minute to knock the French side out. “We are talking about players who’ve won European Cups, World Cups, but when they lose confidence they went out.” No chickens are being counted here.

Swansea’s position, still in the bottom three, is more perilous. As their manager, Paul Clement, said, their best news of the day came from elsewhere, with Middlesbrough drawing and Hull losing to Manchester City.

“There was a lot of fear in our play – I can’t sit here and make excuses for that. It wasn’t good enough,” said Clement, pointing to the physically and psychologically draining late defeat to Tottenham as a possible cause for today’s struggles. “We put in a massive shift on Wednesday and there was a lot of heartbreak at the end of the game.”

A Cheikhou Kouyaté goal just before half-time was enough to snag the three points for West Ham, though it was a game that virtually nobody could have enjoyed. Not the home team, who had to suffer horrific tension, not the away team, who lost, and not neutrals, who were hardly provided with an afternoon of rip-roaring entertainment.

The opening stages were played as if both teams had been challenged not to make moves of anything over three or four passes at a time – a bit like a large-scale and very expensive version of the Crystal Maze, only with less skill and in less atmospheric surroundings.

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The first real moment of any quality came from the Swansea keeper, Lukasz Fabianski, who, while falling back into his own goal managed to somehow contort his body in such a way as to stop a Robert Snodgrass header from going past him. The home fans, desperate for some – any – good news, pleaded for the referee’s watch to buzz with conformation that the ball had crossed the line, but play continued.

After about half an hour Clement looked at his own watch. Maybe he was bored. Fabianski produced another decent save to stop André Ayew’s low shot, but the general level of quality remained low, to put it kindly. This looked like a game between two sides struggling at the grim end of the table with the prospect of relegation looming. Which perhaps should have been no particular surprise.

West Ham took the lead just before half-time after a neat passing interchange – one of the first of the afternoon – fed Kouyaté outside the area, and his powerful shot dipped into the bottom corner. The midfielder dashed into the waiting arms of the home fans, who hugged him tightly: for this spontaneous expression of relief and joy he was, of course, booked.

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Having the lead didn’t seem to calm Bilic’s nerves. He limped around the London Stadium’s vast technical area, hands on hips, pausing only occasionally to move the hands to his knees and double over like a man with chronic stomach cramps. “It is not a time to panic or make kneejerk decisions,” wrote the co-owner David Sullivan in his programme notes. “We have 100% faith in Slaven Bilic’s ability to lead West Ham United forward.” That reassurance had not seemed to relax the Hammers manager.

As time ticked down Swansea inched their way into the game, missing a couple of presentable chances, while at the other end Ayew and Jonathan Calleri could have made things less tense. In the end for West Ham, it did not matter.