Westwood started to lose his grip on the jug with bogeys on the seventh and eighth, and failing to birdie the downwind, par-5 ninth. Presented with birdie chances early on the back nine, his putting stroke began to look tentative. He hit into the dunes on the right side of the 13th to make bogey and never caught up.
Westwood and Scott tied for third with Ian Poulter, who played a four-hole stretch in 5-under around the turn and closed with a 67. At 1-over 285, he canceled a flight home in case of a playoff. Moments later, with Mickelson pulling away, the outcome was clear.
Jack Nicklaus said on Twitter, "Phil's round was incredible. After his bad break on 16 and to then get up and down showed a lot of guts. And the two great shots at 17 ended the tournament."
Mickelson had to wait for four groups to finish before he could claim the oldest championship in golf. He huddled with his wife and three children, who returned from a quick holiday in Spain, and shared a long embrace.
Making this even sweeter for Mickelson is that just one month ago he lost out on yet another chance to win the U.S. Open, the missing link of a career Grand Slam. Mickelson twice made bogey with wedge in his hand on the back nine at Merion and had his record sixth runner-up finish.
Woods, Nicklaus, Gary Player, Ben Hogan and Gene Sarazen are the only players to win all four professional majors. Mickelson figured it would be the British Open that would hold him back. Now he has the jug, and he never took his hand off during his press conference.
"I think that if I'm able to win the U.S. Open and complete the career Grand Slam, I think that that's the sign of the complete, great player," Mickelson said. "And I'm a leg away. And it's been a tough leg for me."
For now, Mickelson takes his place among an elite list of winners at Muirfield, which is considered the fairest of the links on the British Open rotation. All but two of the Open champions at Muirfield are in the World Hall of Fame. Mickelson is the only winner who already has been inducted.
It was the 43rd win of his PGA Tour career. The guy who once couldn't win the big one now has five majors in the last nine years. This one returns him to No. 2 in the world ranking for the first time in nearly three years.
Woods, meanwhile, now has gone 17 majors without winning, and that pursuit of Nicklaus and his benchmark of 18 majors — Woods is stuck on 14 — doesn't look any closer. He three-putted twice in four holes and looked like just another contender on this Sunday.
He attributed his round to not getting the right pace on the greens, which he said were progressively slower.
"I felt like I was really playing well today, actually the whole week, " said Woods, who has not broken 70 in the final round of his last seven majors. "I really hit so many good shots and really had control of my ball this week. As I said, it was just trying to get the speed, and I just didn't get it."
Woods tied for sixth with Zach Johnson (72) and Hideki Matsuyama of Japan, who was penalized one shot on Saturday for slow play. The 21-year-old Matsuyama, who twice made the cut at the Masters as a teenage amateur, closed with a 70. He has finished in the top 10 in the U.S. Open and British Open this summer.
Hunter Mahan, playing in the final group in back-to-back majors, made a short eagle putt on the ninth to get within one shot of the lead. He bogeyed two of the next three holes and shot 75.