The team shot four birdies and one bogey to pull ahead in the alternate-shot fourth round, having begun the day tied for the lead with Scott Stallings and Trey Mullinax.
A birdie putt from Palmer on the par-5 second hole gave them the edge they needed and the duo remained in front for the rest of the round - with Rahm draining a 24-footer to save par on the 15th and continue the momentum.
The pair took a three-shot victory over Sergio Garcia and Tommy Fleetwood, finishing with a total of 26-under 262, while Stallings and Mullinax tumbled down the leaderboard with four bogeys, a double bogey and one birdie for a final round of 77.
Rahm said the win had been a team effort.
Speaking during a press conference, he said: "It's been incredible. The way it happened - he started off playing great, I kind of struggled and then I came through in the middle, and then he finished it off - it was amazing.
"To win as a team, it's something you don't get to feel very often in golf. I hope it happens again."
Garcia and Fleetwood had an eagle, four birdies and two bogeys to card a final round of 68 and finish with a total of 23-under 265, while Kyoung-Hoon Lee and Matt Every finished two shots further back to tie for third alongside Brian Gay and Rory Sabbatini.
For teams tied for fifth on 20-under.