Rodriguez's solo goal gives Southampton a hard-fought 1-0 victory over Crystal Palace at Selhurst Park.
Scott Brown scored his second Scotland winner in two games as Gordon Strachan's side beat Poland 1-0 in a Warsaw friendly.
Liverpool moved into second place with a 3-0 victory at Southampton.
West Ham were dominated for long periods but somehow beat Southampton 3-1 to climb further away from the Premier League drop zone.
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