The outcome denied Sunderland manager David Moyes his first taste of success at the club which was seemingly within his grasp when Defoe hammered home a penalty after he was brought down by Jose Fonte.
But Rodriguez, thrown on by Southampton's French manager Claude Puel in the 74th minute for the ineffective Charlie Austin, beat Jordan Pickford with a tame-looking effort from 25 metres which somehow squeezed under the visiting keeper.
It was a cruel end for Pickford who had kept his team afloat with a string of good saves, notably when he denied Austin and Shane Long either side of halftime.
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(Writing by Zoran Milosavljevic; Editing by Ed Osmond)
