World number one and defending champion Brooks Koepka withdrew with a recurrence of a long-standing knee injury.
South Korea's An, who shot 69, shares second on 11-under going into the weekend.
Second-ranked McIlroy's last major victory came at the 2014 US PGA Championship, while Koepka was still playing on the European Tour and yet to take up his PGA Tour card.
Despite winning his first major, Woodland missed out on automatic selection for the Ryder Cup-style event that will pit Team USA against the Internationals in Melbourne in December.
The 34-year-old had started the day three shots behind Matthew Fitzpatrick but a bogey-free final round of 65 saw him finish on 16-under for the tournament, beating the Englishman by one shot.
Malnati, who admitted that he's been distracted with his wife, Alicia, expecting their first child this month, buckled down to post a seven-under par 65 that gave him a 36-hole total of 10-under 134.
By the 15th, Rahm had clinched four birdies in a row, with an incredible second shot at the 16th spinning in for an eagle.
The 27-year-old recorded a final round of 70 at the Old Course at St Andrews to reach 22 under par and become the first Frenchman to win the tournament.
Each of them will make four captain's picks in early November to complete his 12-man squad for the biennial team golf showdown in December at Australia's famed Royal Melbourne layout.
Niemann had three birdies and his first bogey of the week to emerge with a 15-under par total of 195 on the Old White TPC in White Sulphur Springs, West Virginia.
Northern Ireland star McIlroy, who was also handed the accolade in 2012 and 2014, came out on top after a vote by fellow tour professionals.
Rookie MacIntyre, 23, had taken a four-shot lead to grab the lead from Englishman Paul Casey on Friday but could not hold his advantage in a finely poised battle at the Green Eagle Golf Course.
The four-time major champion made three bogeys in his first 12 holes but stormed home with a four-hole run of three birdies and an eagle.
The European Tour event has been held at the Doha Golf Club course since its inception in 1998.
In a brief statement on Twitter, reigning Masters champion Woods said he had undergone keyhole surgery last week to mend "minor cartilage damage."