The laid-back journeyman, previously a runner-up three times on the U.S. circuit, sealed the long-awaited win in bright sunshine with a two-putt birdie on the second extra hole at the TPC Louisiana, the par-five 18th.
Els, seeking his first PGA Tour victory since the 2010 Arnold Palmer Invitational, missed his long-range birdie attempt there from just off the left edge of the green.
"It was a tough day," Dufner, 35, said in a greenside interview after finally triumphing on his 164th PGA Tour start. "Ernie put up a good number today and we were battling it.
"It was just tough trying to get this first win. It's been tough for about six years out here now, so a little bit of a monkey off my back. I'm excited."
Both players had good birdie chances at the first extra hole, also the 18th, but Dufner missed his attempt there from seven feet and Els from six feet.
The duo had finished the regulation 72 holes on 19-under-par 269, Els eagling the seventh on his way to a five-under 67 and Dufner narrowly missing a 10-foot birdie putt at the last to close with a 70.
Britain's Luke Donald fired a 67 to finish alone in third at 17 under and will regain the world number one spot from Northern Irishman Rory McIlroy when the official rankings are issued on Monday.
Canada's Graham DeLaet (71) and American Ryan Palmer (69) tied for fourth at 16 under, a stroke in front of American world number seven Steve Stricker (70).
Dufner, who played with his customary unflappable demeanour all week, held a two-stroke lead overnight but that advantage was soon trimmed to one after fast-starting American Ken Duke birdied four of the first five holes.
Els also got to 16 under, level with Duke, after knocking in a four-foot birdie putt at the par-four fifth.
While Duke slipped back with a bogey at the eighth and a double-bogey at the ninth, Els found himself in a two-way tie at the top after Dufner bogeyed the par-four sixth.
Els then took control in spectacular style with an eagle at the par-five seventh, hitting his second shot from 228 yards to eight feet and rolling in the putt to forge two strokes clear.
Dufner, however, immediately responded with birdies at the seventh and eighth to draw level with Els at 18 under as the two players duelled for the title.
After safely parring the ninth to reach the turn in one-under 35, Dufner coolly sank a 10-footer to birdie the 10th and regain a one-stroke lead.
Els again caught Dufner at the top with a birdie at the par-five 11th and both players did well to save par at the tricky 14th after overhitting the green with their tee shots.
Dufner also salvaged an unexpected par at the 16th where he drove into the lake bordering the fairway, took an unplayable lie, then struck his third shot to 45 feet before remarkably holing the putt.
As the late afternoon shadows lengthened, both players parred the last two holes to take the tournament into a sudden-death playoff.
Masters champion Bubba Watson, who won last year's New Orleans Classic in a playoff, ended his title defence at the TPC Louisiana with a 70 to share 18th place at 11 under.