Kisner's pitch from 30 yards at the par-five 18th clattered against the pin and dropped in to give the American duo a final round of 60 for a 27-under-par 261 total.
Australian Smith and Swede Blixt shot a closing 64 to also finish four strokes clear of Americans Kevin Tway and Kelly Kraft.
Kisner did not spend much time pondering his eagle shot at the last. He was almost resigned to defeat, trailing Smith and Blixt by one stroke, with Smith facing little more than a tap-in for birdie.
"We knew we had to make it," Kisner, winner of one PGA Tour event, said of his unlikely attempt. "I knew they were going to make birdie, so all I was trying to do was make sure I didn’t leave it short."
Kisner said he "couldn’t see much" in the poor light, and he let out a massive scream of delight when he realised his ball had gone in, before giving Brown a hard high-five.
“When it hit the flag, I said ‘don’t you come out of there’,” he said.
After the crowd settled, Smith made his birdie from little more than a foot.
He was not the first Australian on the receiving end of a final-hole dagger at the New Orleans Classic.
In 1990, Greg Norman was shocked when South African David Frost holed a winning bunker shot at the last, one of several bitter pills The Shark endured.
Blixt admitted he was surprised by Kisner’s eagle.
"Honestly I didn’t really expect that, but that’s part of golf, you’ve got to expect the unexpected so we get another shot at it tomorrow,” said the two-time PGA Tour winner.
"We’ve got to leave this behind and go out there and make birdie or eagle tomorrow.”
Earlier, Kisner and Brown had birdied the first eight holes to turn a four-stroke deficit into a two-shot advantage and they did not fall behind again until Smith and Blixt birdied the penultimate hole.
The final round was played using the four-ball (best ball) format, where each competitor plays his own ball and the team uses the best score on each hole.
(Reporting by Andrew Both in Cary, North Carolina, editing by Nick Mulvenney)
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