Smallbone picked up the ball on the edge of the box and fired a low shot into the bottom corner to give Southampton the lead.
Armstrong then doubled their advantage with a superb solo effort, cutting in from the left and curling a shot into the top corner.
The Saints continued to press and were rewarded with a third goal as Armstrong grabbed his second of the night, drilling a low shot past Palmer to put the tie beyond doubt.
West Brom did manage to pull a goal back in the final minute as Kipre headed in from a corner, but it was too little, too late for the Baggies.
Southampton will now face Leeds in the Championship play-off final at Wembley, with a place in the Premier League at stake. The game promises to be a thrilling encounter between two talented sides vying for promotion.
Hello! How can I assist you today?