They say the man, named as Richard White, 62, attacked an airport security worker with wasp spray before striking another on the arm with a large knife.
Police said he was "unresponsive" after being shot in the leg, chest and face but was taken to a local hospital.
Several bystanders were said to have been treated for minor injuries.
Jefferson Parish Sheriff Newell Normand told reporters it was unclear whether White was at the airport as a traveller. "We're still piecing together witness interviews," he said.
He said White, a taxi driver, had "little or no criminal history". He was alive and in surgery, Sheriff Normand added.
Those making their way to and from flights in the busy terminal reported chaotic scenes as the incident unfolded.
Jeremy Didier, who had just arrived on a flight from Florida, said he saw a man "jumping over people" in the queue at the boarding pass checkpoint.
Two gunshots rang out and "everyone hit the floor," he told The Times-Picayune, a New Orleans newspaper.
Farah Stockman, a journalist for the Boston Globe who was in the airport at the time, tweeted: "Just witnessed [the] craziest thing."
"People had heard gunshots. Everyone was hiding behind chairs and in doorways. Make our way to exit. Body was lying in a pool of blood."
Authorities said the area where the attack took place would be closed off until Saturday but confirmed that rest of the airport remained open.

The Piece Hall in Halifax opened in 1779 as a trading centre for locally-woven cloth.
Calderdale Council leader Tim Swift said the "fascinating finds" hinted at life in the Piece Hall over the past 230 years.
It is undergoing a Â£19m refurbishment scheme and is due to re-open in Spring.
Three photographs were found behind skirting boards of one of the former shops.
One shows a young boy holding what may be a catapult, standing next to a young girl in a hat. Two other Victorian studio portraits are of smartly-dressed gentlemen.
Calderdale Council said one was a tintype photograph, widely used in the 1860s and 70s.
An un-lidded wooden box was found labelled with Emsley & Collins Ltd, possibly a local company, and a pile of oyster shells was found in the cellar.
Mr Swift said: "The fascinating items provide a tantalising view of a forgotten world. It would be great to know the stories behind them but unfortunately we know very little.
"The artefacts have little historic value but they provide a hint at what daily life may have been like in the Piece Hall during the past 230 years."
In the 1970s, the 18th century hall became a tourist attraction with an art galley, museum and shops.
It closed in January 2014 for a two-year renovation and to add an extension and new visitors centre.
The Â£19m project is funded by Calderdale Council, the Heritage Lottery Fund, Garfield Weston Foundation and the Wolfson Foundation.

The blaze broke out at Coed Eva Primary School, in Cwmbran, at about 02:25 GMT on New Year's Day.
Most pupils returned on Monday, though nursery children will have to wait until 25 January when temporary classrooms will have been installed.
Six people, including four teenage boys, arrested in connection with the fire have been released on bailed.

The American world number one shot seven birdies and an eagle as he moved to 16 under and overtook Patrick Reed.
Reed (69) is joined on 12 under by fellow American Kevin Kisner (65) and Argentina's Fabian Gomez (66).
Irishman Padraig Harrington (68) is eight under and Australia's US PGA champion Jason Day is three under (73), alongside Scotland's Russell Knox (72).
Northern Ireland's Graeme McDowell is one under after a 70 at the Plantation Course in Kapalua.
Masters and US Open champion Spieth, who is bogey-free after his first two rounds, said: "It is nice to have no blemishes on the card. I want to prepare and start the year with a bang, but I'm only through 36 holes. There's a long way to go."
Rickie Fowler (67) is on 10 under, while two-time Masters champion Bubba Watson (68) is a shot further back.
US security staff shot a knife-wielding man three times at New Orleans international airport after he tried to storm a checkpoint, officials say.
[NEXT_CONCEPT]
Old photos, oyster shells and a World War Two booklet have been found by workmen renovating a Grade I listed building in West Yorkshire.
[NEXT_CONCEPT]
An infants and nursery school in Torfaen has reopened 10 days after it was gutted by fire.
[NEXT_CONCEPT]
Jordan Spieth leads the Tournament of Champions by four shots after shooting a nine-under second-round 64 in Hawaii.