An 83rd-minute James McClean own goal appeared to have earned the Gunners a hard-fought away win at The Hawthorns until Rodriguez levelled from the spot in the 89th minute, following a handball by Calum Chambers.
In the first half Arsenal's Alex Iwobi curled over before Baggies goalkeeper Ben Foster saved two long-range shots.
Though Rodriguez forced a good save from Petr Cech, who was aiming for a record 200th PL clean sheet, Arsenal looked to have sealed victory when Alexis Sanchez’s free-kick deflected in off McClean.
But West Brom ensured they remain undefeated against the Gunners in their last three home matches, claiming a point that takes them 19th. Arsenal lie fifth.
West Brom: 2 Jan v West Ham (A)
Arsenal: 3 Jan v Chelsea (H)