Yoshida broke the deadlock with a predatory finish early on in the game, which the hosts looked comfortable in until Victor Wanyama was sent off shortly into the second half of the contest.
The Hammers threw strikers Andy Carroll and Emmanuel Emenike on to try for a leveller, but a dogged performance from Saints was rewarded by a fifth clean sheet in a row to stretch their winning league run at St Mary’s to a fourth match.
Ronald Koeman made three changes to the team that started Tuesday's draw at Arsenal with Yoshida joining Jordy Clasie and Graziano Pellè in coming into the team. Striker Charlie Austin also returned to the squad ahead of his St Mary’s bow.
Saints got the game under way attacking the Chapel stand, and kept the Hammers quiet early on before venturing forward.
On eight minutes Shane Long delivered a cross from the left which Cédric headed on target as he came in from the other flank on the run, only to be denied by Hammers 'keeper Adrian.
A minute later Saints took the lead with a strike that owed to perseverance and instinct as Sadio Mané and Wanyama combined on the left, with the latter forcing the ball into the centre of the area for Yoshida to smash the ball home from close range for 1-0.
Koeman’s men continued to have the better of the chances. A 25th-minute free-kick from Ryan Bertrand found Pellè inside the area, but an unfortunate bounce and a quick clearance prevented both the Italian forward and a late-arriving José Fonte from getting another touch of the ball.
The Hammers responded with a couple of chances either side of the half-hour mark, with Fraser Forster called upon to deny Winston Reid’s header from a Dimitri Payet corner before the same combination led to the ball flying over the target at the near post.
Saints held their one-goal lead at the break, and there were no changes from either side at the start of the second half.
However, shortly after Forster had been called into action to deny a low shot from Enner Valencia, Saints were dealt a blow when Wanyama was sent off for a challenge from behind on Payet.
The sending-off forced Koeman to shuffle his pack, with Oriol Romeu replacing Mané. The Hammers then introduced Sam Byram for the injured James Tomkins at the same time as Andy Carroll’s entrance onto the pitch in place of Victor Moses.
Saints didn’t sit back despite their one-man disadvantage, and created an opening on 63 minutes when a long ball forward fell kindly to Long who couldn’t keep his effort down as he bore down on goal.
Pellè made way for Austin for the final 20 minutes or so, as Saints opted for two frontmen to trouble the Hammers defence in the closing stages.
West Ham started to create chances for themselves at the other end as the game opened up, with Song curling a shot wide before substitute Emenike headed wide.
A fantastic sweeping ball forward from Romeu then released Austin into the final third where the striker won a corner from which Clasie, at the second attempt, tested Hammers ‘keeper Adrián from range.
The last ten minutes saw Saints contend with a barrage from the visitors, though their efforts were mainly from range as Forster remained largely untroubled.
Saints had a late chance to wrap up the win when Romeu twisted and turned his way through midfield before laying the ball into Austin’s path, but the striker’s shot just squirmed wide.
At the other end, Cédric came close to slicing the ball into his own net after as he tried to clear the ball into the wind prior to five minutes of stoppage time added on to test the hosts’ patience.
But they held on with a resilient display late on to record a crucial win against the team one place above them in the table.
Southampton team (3-4-2-1): Fraser Forster; Virgil van Dijk, José Fonte, Maya Yoshida; Cédric, Victor Wanyama, Jordy Clasie (James Ward-Prowse 90), Ryan Bertrand; Sadio Mané (Oriol Romeu 56), Shane Long; Graziano Pellè (Charlie Austin 72). Unused substitutes: Maarten Stekelenburg (GK), Dušan Tadić, Cuco Martina, Juanmi.
Sent off: Wanyama (54)
West Ham team (4-3-3): Adrián; James Tomkins (Sam Byram 61), Winston Reid, James Collins, Aaron Cresswell; Alex Song, Mark Noble, Dimitri Payet; Michail Antonio (Emmanuel Emenike 73), Victor Moses (Andy Carroll 61), Enner Valencia. Unused substitutes: Darren Randolph (GK), Pedro Obiang, Joey O’Brien, Reece Oxford,
Booked: Valencia (67), Byram (87)
Referee: Mark Clattenburg
Attendance: 29,161 (2,601 away fans)