The Gunners knew only a win would lift them back into the Champions League places after Tottenham beat Leicester earlier in the afternoon, but Mikel Arteta's side responded by grinding out an important three points.
It was Arteta's centre-backs who came up with the goods, with Gabriel Magalhaes nodding the winner early in the second half after Rob Holding had opened the scoring from Bukayo Saka's 38th-minute corner.
"We didn't play so well," Arteta said.
"If I have to use a word it would be we won ugly. Big teams have to find a way to win ugly when they don't play well."
Speaking on Sky Sports, he added: "We found a way through set pieces and defending strongly. To win here at West Ham is credit to boys and shows the spirit we have."
Jarrod Bowen did pull the Hammers level just before the break, controlling Vladimir Coufal's cross and squeezing a strike into the bottom corner in what was a reward for a strong first-half performance.
But David Moyes' outfit failed to reach those levels after going behind for a second time - and defeat leaves them seventh, three points behind sixth-placed Manchester United with three games remaining.
"I thought we had done well enough," Moyes said.
"We conceded a couple of set pieces which we are normally very good at, both attacking and defending. We also made a few changes which left out some of the bigger players who might have been involved in those situations.
"We knew that was a risk and we got punished for it."
The Gunners are desperate to return to Europe's elite competition next season and Arteta's side have enjoyed the advantage of having no continental football this term, meaning all their resources can be focused on achieving a top-four finish for the first time since 2015-16.
West Ham's only realistic chance of qualifying for the Champions League now is to win the Europa League - and Moyes' team would be forgiven for having one eye on overturning their 2-1 deficit in the semi-final second leg against Eintracht Frankfurt in Germany on Thursday.
Arsenal struggled to get a foothold, though, before Eddie Nketiah's clever run and low shot forced a corner, from which Holding climbed highest to head in Saka's delivery for his first Premier League goal.
Bowen's smart finish pulled the Hammers level on the brink of half-time. But, after the hosts failed to properly clear a corner, Gabriel was left unmarked at the back post to head in Gabriel Martinelli's 54th-minute cross, beating former Arsenal stopper Lukasz Fabianski at his near post.
Nketiah could have sealed the success on the break, only to curl wide, but the 22-year-old continued to prove a threat after again being trusted to lead the line for Arsenal.
The visitors were not at their fluent best, but as the battle with north London rivals Spurs hots up - and with the Gunners heading to Tottenham a week on Thursday - restoring their two-point advantage was the most important outcome for Arsenal.
Injuries and suspensions have hit the Hammers and they found themselves without three senior centre-backs in Craig Dawson, Issa Diop and Angelo Ogbonna on Sunday, with left-back Aaron Cresswell chosen to partner Kurt Zouma.
That may have been partly to blame for the hosts coming unstuck twice from set-pieces, but this was also West Ham's 52nd match of the season - 11 more than Arsenal have played - and they struggled to unpick the Gunners as the game wore on.
The hosts found themselves behind despite having the better of the opening half an hour. They responded to Holding's opener within seven minutes when Bowen grabbed his 10th top-flight goal of the campaign, becoming the first Englishman to reach double figures for the club since Kevin Nolan in 2012-13.
Moyes turned to Michail Antonio, rested before Thursday's Europa League semi-final second leg, with 19 minutes remaining to try to find a leveller for the second time, but the forward only managed one touch in the opposition box.
The Hammers have now dropped points in each of their past four Premier League matches following a Europa League fixture, while this loss brought their six-game unbeaten home run in the league to an end.
More worryingly for Moyes, his side are winless in their past four Premier League outings - and the Scot will not want what has been a productive season to fizzle out now.
