Long-term leaders Jonas Blixt and Cameron Smith took a four-shot lead into the final round of fourballs.
But Kisner and Brown birdied 10 of the first 11 holes and Kisner then chipped in for an eagle at the last.
Smith had to birdie the last to join them at 27 under and the four men will return for a sudden-death play-off.
Australian world number 112 Smith holed from two feet for the closing birdie, but darkness following a delay of more than six hours because of thunderstorms at TPC Louisiana midway through the round meant the play-off could not take place on Sunday.
Sweden's Blixt - the world number 256 - and Smith had led outright after the second and third rounds and did not drop a shot throughout the 72 holes, with each round in the revamped team format alternating between foursomes and fourballs.
The final tee time was moved forward to 8:16am local time (14:16 BST) in an attempt to beat the predicted bad weather and despite the intense thunderstorm, it appeared the tournament - the first official PGA team event since 1981 - would be completed.
But Kisner's 31-yard chip, close to 12 hours after his group had teed off, rattled the pin and dropped into the cup, meaning the players will return to the par-five 18th to begin the play-off at 9am local time (15:00 BST) on Monday.
"Well, we knew we had to have it," Kisner said. "All I was trying to do was make sure I didn't leave it short, and I couldn't see much. I knew it was breaking a little right, and when it hit the flag, I said, 'Don't you come out of there.'
"Someone had to get hot in order to catch us today, and they did," said Blixt, a two-time PGA Tour winner. "We just have to leave this behind and try to go out there and make birdie or eagle on 18 tomorrow and try and win this tournament," he added of the better-ball play-off.
Americans Kelly Kraft and Kevin Tway were third after combining for a 61 that left them on 23 under.
Former world number one Jordan Spieth and fellow Texan Ryan Palmer were a further stroke back after posting a 64.
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Americans Kevin Kisner and Scott Brown carded a 12-under final round of 60 to take the Classic of New Orleans team event into a play-off on Monday.