With the victory, Rahm joined Seve Ballesteros as the only Spaniards to reach No. 1 in the world. It was the 10th win of Rahm's career and his fourth on the PGA Tour.
Rahm closed with a 3-over 75 for a 3-shot win over Palmer, which was the highest final round by a Memorial winner since Roger Maltbie shot 76 in the 1976 tournament.
Rahm built an eight-shot lead at the turn but saw it reduced to three strokes with three holes left to play. The Spaniard wrapped up his victory with a magical flop shot on hole No. 16 from deep grass that landed on the fringe and rolled toward the hole, dropping in for birdie.
PGA Tour officials reviewed Rahm's spectacular shot and gave him a two-shot penalty. The replay showed his lob wedge slightly moved the ball before the shot, but the penalty didn't impact the outcome.
"It doesn't take anything from the shot," Rahm said. "It's not going to take anything away from today."
Following his victory, Jack Nicklaus greeted Rahm behind the 18th green with a fist bump. Earlier Sunday, Nicklaus revealed that he and his wife, Barbara, tested positive for the coronavirus in March.
Nicklaus, who has since tested positive for antibodies, said he favored a handshake -- a tradition at the event he created -- but left the decision up to the winner.
Tiger Woods, who was playing in his first competition since February, shot a 4-over 76 in the final round to finish 6-over par and tied for 40th. It was his third-worst total score relative to par at the Memorial in 18 career appearances.
"These were some of the more difficult conditions I've played in a long time," Woods said. "[Sunday] the golf course got quick and got fast. It was hard to get the ball close, and there weren't -- I don't think there will be too many low scores [Sunday]."
Matthew Fitzpatrick had a 68 for the lowest score of the final day to finish third. Jason Day and Matt Wallace tied for fourth at 4-under 284, while Mackenzie Hughes and Henrik Norlander finished at 3-under 285 for a share of sixth.