At the end of a week in which an England match was postponed because no one in Warsaw could close a roof to prevent rain from flooding the national pitch, Nigel Adkins turned to a Polish stopper to plug Southampton's leaky defence. So Artur Boruc, the former Celtic goalkeeper signed as a free agent in late September, made his Premier League debut – but could not deter another deluge of goals. A slew of second-half errors means the Saints have now shipped an average of three goals per game this season.

Continue like this and Southampton will be sunk from the Premier League. And Adkins could be cut adrift by his employers. By contrast, everything is going swimmingly for West Ham, who finished behind Southampton in last season's Championship but are currently looking far more at home in the top flight. That said, the first half here was sub-Championship standard, so much so that even Sam Allardyce admitted afterwards: "I was surprised to win 4-1 after such a lacklustre first half."

Boruc's first touch set the tone: a botched kick by the goalkeeper in the fifth minute presented the ball straight to Andy Carroll. The striker's instinct looked rusty, as he dithered before attempting a 20-yard shot that José Fonte blocked easily.

Carroll was the focus of most of West Ham's early play, with the team's midfielders lofting the ball towards him from wide and deep, like golfers aiming for a green's flagpole from all over the fairway. Southampton coped well enough with the aerial bombardment but looked less comfortable trying to stop Matt Jarvis, whose trickery down the left was an intermittent irritant. One feint, dash and cross by the winger in the 15th minute resulted in Jos Hooiveld having to clear a James Tomkins header off the line.

West Ham just about had the upper hand despite the decision by Adkins to deploy a five-man midfield, presumably in an effort to control possession.

The closest Southampton came to bothering Jussi Jaaskelainen in the first half was when Jay Rodriguez took down a perceptive pass from Morgan Schneiderlin before stabbing wide from 18 yards. It was not a bad attempt but the miss inevitably prompted wondering as to whether Rickie Lambert would have done better — Adkins had left Southampton's top scorer on the bench, as he did at Manchester City on the first day of the season. The ploy did not work on that occasion and it did not work here.

"They've conceded a lot of goals and are struggling confidence-wise so I said that if we put pressure on them they will make mistakes," said Allardyce of his half-time team-talk. He was quickly proved right.

Within a minute of the resumption the deadlock was broken in suitably scruffy fashion. Mark Noble clipped a 40-yard free-kick into the box; Fonte and James Collins challenged for it but neither connected and the ball bounced past Boruc and into the corner of the net. Southampton's prospects of a first clean sheet of the season were in tatters. So was their defence a minute later when the left-back Maya Yoshida was beaten to a breaking ball by Yossi Benayoun. The Israeli, making his second debut for the club following his summer transfer, scampered into the box before sliding low pass across the face of goal to Kevin Nolan, who tapped into the net.

Southampton nearly allowed Nolan to inflict more damage in the 60th minute, when he got on the end of the quickly-taken freekick, only for his close-range effort to be blocked by Boruc. Two minutes later Southampton hit back in style. Rodriguez seemed to be attempting a shot when he slipped and spooned the ball to Adam Lallana, who swivelled near the penalty spot before shooting into the roof of the net.

Lambert was introduced one minute later and almost made an immediate impact, curling a 20-yard free-kick just over. But then Southampton, just as they seemed to be working up a head of steam, blundered at the back again, Fonte handling the ball while jumping for it with Carroll. Noble sent Boruc the wrong way from the spot.

Moments after replacing Carroll in the 81st minute, Modibo Maïga took advantage of more feeble defending by Southampton to wriggle into the box and fire into the corner of the net. "We conceded very poor goals today," admitted Adkins.

Asked how long he expects to be given to sort out this persistent problem, he replied: "Honest answer? I don't know."