The world number 324, who represented Australia until 2013, swept through the first set against the Argentine.
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The 27-year-old Briton then wasted four match points in the second set tie-break as Bagnis hit back to level.
But Klein retained his composure and overturned a 5-2 deficit in the final set to win the match in the tie-break.
Klein will play Portugal's Joao Sousa in the second round.
Briton Brydan Klein reached the second round of the Auckland Open with a 6-1 6-7 (6-8) 7-6 (7-3) win over Facundo Bagnis - ranked 269 places above him.