Alexandre Lacazette tested West Ham goalkeeper Lukasz Fabianski early on, but the Hammers soon took control with Samir Nasri pulling a shot wide.
Rice headed wide from a corner just before half-time, but the 19-year-old made no mistake three minutes after the break.
Anderson's cross was half-cleared and Nasri teed up Rice to curl into the top corner.
Alex Iwobi and Pierre-Emerick Aubameyang both went close to equalising but the Hammers held firm.
West Ham climb two places to eighth on 31 points, while Arsenal stay fifth on 41 points.
West Ham: 19 Jan v AFC Bournemouth (A)
Arsenal: 19 Jan v Chelsea (H)