The win over Saints not only lifted West Ham clear of the drop zone, it also lifted the mood around the whole club following the protests and pitch invasions which marred their previous match against Burnley.
Chelsea took the lead from an unlikely source as Cesar Azpilicueta scored from a corner.
Chelsea remain fifth on 57 points and are now 10 points behind fourth-placed Tottenham Hotspur in the league table.
Morata failed to test Hart with a towering header that soared over the target on 55 minutes, while Moses curled a left-footed shot narrowly wide of the left post 14 minutes later after Masuaku lost possession inside his box.More news: Puigdemont urges Spain to open negotiations over Catalonia
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And even though that performance will have done his chances of securing a place in Gareth Southgate's squad, if not starting XI, no harm, Hart was adamant his focus sits exclusively on helping West Ham maintain their Premier League status.
Willian had a fierce drive diverted behind before Morata saw a goal ruled out after straying into an offside position to receive Hazard's incisive pass.
ExWHUemployee, a solid source of West Ham news, reported that the player would not make the game on Sunday.
It was end to end action in the latter stages but Hart's resilience kept Chelsea at bay, thankfully our defence did just about enough to stop David Moyes' side from getting the victor.
But Cairney's visit to Stamford Bridge is sure to only increase speculation that he could move to the London Stadium, ahead of the summer transfer window.
"I'm not disappointed. I thought we played against a really slick Chelsea team, particularly in the first half, and I was happy to get in only 1-0 down".