The pair had finished tied on 17 under par after Mickelson three-putted the 72nd hole for a bogey six, charging his birdie putt four feet past the hole and missing the return.
But the world number eight made amends when he returned to the 18th for the first extra hole, hitting a trademark pitch which span back to within inches of the hole for a birdie that Grace could not match.
"This is really cool," said Mickelson, who claimed his first individual victory in Britain in the first regular European Tour event being screened live over the weekend in the United States.
"I have been coming here for some time and had some opportunities and I almost let it slip away today. To come out on top was terrific. It's very special for me to have some success over here.
"It's been the biggest challenge of my career, adapting to links-style golf and this was a great challenge in the final round in some difficult conditions. I played some good golf to come out on top and this is really fulfilling and special.
"I was so mad at myself for mentally losing my focus that I came out on that 18th pretty focused to make a four. (But) nobody likes a movie that's predictable. You always want to have a little bit of suspense."
Asked if it was possible to win back-to-back events and lift the Claret Jug for the first time next week, Mickelson added: "I certainly hope so. I don't think there is a better way to get ready for a major championship than playing the week before and getting into contention and coming out on top just gives me more confidence.
"But there's a lot of luck involved in the Open. You need some luck with your tee times, you need to play well and have some good bounces.
"This really was valuable (preparation), not only for the things I did well but because I have a few things I need to work on. Today was a great day to have this type of weather because the first three days were very benign. I am anticipating this type of weather next week and if we get it, this week is going to be very important."
Mickelson began the final round two shots behind playing partner Henrik Stenson and ran up a double-bogey six on the first after topping his second shot and three-putting, but birdied four of his next five holes and also picked up three more shots at the start of the back nine.
The last of those on the 14th gave him the lead for the first time and three straight pars, including an excellent save on the 17th, had seemingly secured the £500,000 first prize.
However, the four-time major winner's closing bogey meant a round of 69 to match that of Grace in the group ahead and a second Scottish Open play-off for the left-hander, who lost out to France's Gregory Havret at Loch Lomond in 2007.
Grace had played the back nine in 32 but was unable to birdie the 18th in regulation and was unlucky in the play-off to see his approach pull up around 20ft short of the flag.
"I hit what I thought was a great third shot and it came up short," he said. "I was surprised it didn't skip up coming out of the rough.
"You never know what's going to happen but with the touch he's got you think he's going to do something spectacular and that's what he did, so all the best to him."
Overnight leader Stenson and Denmark's JB Hansen shared third place on 15 under, Hansen having threatened to pull off one of the most remarkable victories ever seen.
The 22-year-old began the final round just two shots off the lead but his chances of a maiden European Tour title looked to have disappeared with a quadruple-bogey nine on the par-five second.
However, Hansen bounced back with five birdies in succession, picked up another shot on the ninth to be out in 34 and when he birdied the 10th as well was suddenly in the outright lead.
The Challenge Tour graduate also got back into a share of the lead after Stenson had overhauled him with birdies on the 10th and 11th, only to drop three shots in his last four holes.
"I'm very disappointed," said Hansen, who started the week with earnings of just £37,000 this season but picked up £195,000 today. "To finish three over for the last four holes is not what I expected.
"I felt really good when I had the lead and had good hold of my nerves. I really had the feeling I could win this. It's been a very valuable experience. I played fantastic golf this week, just what I needed at this time of the season."
A place in the Open was on offer to the winner of the tournament if not already exempt, but with Mickelson taking the title first reserve Scott Stallings gets the Muirfield berth.