Matthew and Han finished regulation knotted at 10-under-par 278 and headed to No. 18 to decide the championship.
Both players found the fairway off the tee at No. 18, the first playoff hole at Tartan Fields Golf Club. Han hit a 5-wood 60 feet from the hole, while Matthew played a rescue club to 15 feet.
Han left herself with 5 feet for par. Matthew lagged her birdie putt to tap-in range, then watched as Han took her chance to extend the playoff. Han missed her putt left and Matthew walked off with the victory.
'I don't think it's quite sunk in yet,' said Matthew, who pocketed $165,000 for the win. 'I played well the last few years, but haven't managed to win. To hold it together and come out with the win, it's fantastic.'
Sunday's final round was a wild ride, that included several lead changes, but in the end it was Matthew and Han, who held the overnight advantage.
Matthew shot a final-round, 4-under 68, while Han managed a 2-under 70.
Lorena Ochoa shot a 5-under 67 to miss the playoff by a stroke at minus-9. Patricia Meunier-Lebouc held a piece of the lead, but a double bogey at 17 and a bogey at 18 dropped her down the leaderboard. She shot a 4-under 68 and tied for fourth with Nancy Scranton, who carded a 70, at minus-7.
Amateur sensation Michelle Wie fired a 3-under 69 and tied for sixth with U.S. Women's Open champion Meg Mallon (70), Mi Hyun Kim (72) and Marilyn Lovander (72). That group came in at 6-under-par 282.
Matthew was only 2 under par on her round and trailed Han and Meunier-Lebouc by two. Han reached 12 under par with a 12-foot birdie putt at the 12th and held a three-stroke lead over Matthew.
Han hit a 9-iron into the right bunker with her third shot at the par-5 14th. She could not get up and down, and her lead over Matthew fell to two.
Meunier-Lebouc, playing several groups ahead of Matthew and Han, was at 10 under, but ran into serious problems down the stretch. Her 9-iron at 17 hit a slope in front of the green and bounced back into the water. That led to double bogey and another approach in the water at 18 dropped her down the leaderboard for good.
That left Matthew and Han on the course with Ochoa in at minus-9. Matthew used a 5-iron at the par-3 15th and stopped it 3 feet from the hole. She made that putt, then hit a wedge to 12 feet to set up birdie at the par-3 17th.
The duo was tied at 10 under par on 18 and Han barely made the sudden-death playoff. Her 4-footer for par snuck in the right side of the cup, but her luck ran out in the playoff.
'My speed was pretty good, but I missed a little on the right side,' said Han, who defeated Wendy Ward in a playoff last year.
For Matthew, this was her second LPGA Tour victory, with her first coming at the 2001 Cup Noodles Hawaiian Ladies Open. She never participated in an LPGA Tour playoff before Sunday and the native of Scotland and the two-time former Solheim Cupper might be more appreciated by mainstream fans of the tour.
'Not so much appreciated, but I haven't won as much,' said Matthew. 'But it was great to win this week. I was trying to get myself ready for it. I'm just absolutely delighted.'