Dufner carded a 2-under-par 70 and Els countered with 5-under 67 in the final round to finish knotted at 19-under-par 269.
The first extra hole came on the par-5 18th at TPC Louisiana.
Dufner was on the green 73 feet from the pin in two shots, but he three-putted to settle for par. Els also parred the hole after finding sand with his second shot.
The pair played No. 18 again for the second playoff hole, and this time the South African two-putted for a birdie while Els missed an 18-footer for birdie from the fringe.
Sharing fourth place were Ryan Palmer (69) and Graham DeLaet (71) at 16-under 272.