Kim and Putnam, neither of whom has won on the PGA Tour, shot three-under 69 in the unforgiving alternate shot foursomes format.
They posted a 13-under 131 halfway total for a one-stroke edge over fellow Americans Kevin Kisner and Scott Brown.
Kim and Putnam had a double-bogey, at the par-four fifth, but that was their only blemish.
"I hit it into places that Andrew had probably never seen before but we grinded it out and pretty pleased with the score today," said Kim.
Putnam, who arrived in the 'Big Easy' in good form after a tie for eighth at the Texas Open last week, added: "I think we did a good job of communicating and scored really well."
Spieth and Palmer, however, seemed to be cruising towards the weekend until it all went wrong at the final two holes.
Spieth pulled his tee shot into a water hazard at the par-three 17th, and then found water again with the pair's second shot at the par-five 18th.
After taking a penalty stroke, Palmer then compounded the misery with a wedge shot that came up short in a bunker, where it plugged, leaving Spieth with an impossible shot that he failed to extricate from the sand.
They shot 74 for four-under 140, missing the cut by one stroke.
First round co-leaders Zhang Xinjun and Dou Zecheng of China also bowed out when they followed an opening 60 in the easier four-ball format with an 80.
Other prominent duos to miss the cut included Sergio Garcia and Rafa Cabrera Bello, Justin Thomas and Bud Cauley, Jon Rahm and Wesley Bryan, and defending champions Cameron Smith and Jonas Blixt.
There were two holes-in-one on Friday. Englishman Chris Paisley used a five-iron from 221 yards at the third hole, while American Billy Hurley III used the same club from 223 yards at the 17th.
The more forgiving four ball (best ball) format will be used on Saturday, before foursomes are played again on Sunday.
(Reporting by Andrew Both in Cary, North Carolina; Editing by Ian Ransom)
