The victory in Saturday's early kick-off saw the Hammers move up to eighth in the Premier League table, on 31 points, ahead of the bulk of the weekend's fixtures.
Irish teenager Rice gave himself an early present for his 20th birthday on Monday by scoring the only goal of the game three minutes into the second half.
The defeat dented Arsenal's bid to break into the top four, with the Gunners still adrift of Chelsea -- who face strugglers Newcastle at Stamford Bridge later Saturday -- in the race for a Champions League place.
That all changed in the 48th minute.
Mustafi's poor clearing header fell to Anderson, whose ball came to Rice via a neat lay-off from Nasri and Rice's first-time shot gave Arsenal goalkeeper Bernd Leno little chance.
Arsenal pressed for an equaliser but the Hammers survived three minutes of stoppage time as they held on for victory.
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