Chelsea first went close at Dagenham when defender Joseph Colley hit the crossbar with a long-range effort.
The decisive goal came on 19 minutes when Kyle Scott capitalised on a poor kick from West Ham goalkeeper Nathan Trott to set up St Clair (pictured above, second left).
Martell Taylor-Crossdale had a goal disallowed as Chelsea pushed for a second.
The hosts went close to an equaliser when Jahmal Hector-Ingram’s deflected effort clipped a post late on.
Chelsea move up to sixth with their fourth victory in five PL2 matches, a point and a place above West Ham.
West Ham: 9 Apr v Everton (A) PL2
Chelsea: 17 Mar v Arsenal (A) PL2