Snedeker closed with a 3-under 69, while Price got into the playoff with a 6-under 66 on Sunday at Midland Country Club. They finished at 16-under-par 272.
The duo returned to the 18th tee for the extra session. Snedeker found the right rough off the tee, but recovered by hitting the green with his approach.
Price, an Australian, landed in the short grass with his tee shot, then dropped his second shot on the putting surface. He attempted his birdie putt first and missed.
Snedeker ran home his 15-foot birdie putt for his second win of the season. He also claimed the Scholarship America Showdown earlier in the season.
After missing several weeks with a broken collarbone, this win virtually secures Snedeker his PGA TOUR card for next season as the $85,500 first-place check moves him to eighth on the money list.
At his win at the Scholarship America Showdown, Snedeker eagled the final hole to force a playoff with Jeff Quinney. He also eagled the last at the Chattanooga Classic to force an extra session, which he lost to Kyle Reifers.
'It was nice to pull from the memories of playing well on the last hole in tournaments earlier this season,' admitted Snedeker. 'I knew I could do it. That obviously helped me today. It was a great way to finish and obviously a good way to finish the playoff.'
Craig Bowden fell one stroke short of the playoff as he fired a 6-under 66 to finish at 15-under-par 273.
Matt Kuchar and Andrew Buckle shared fourth place at 13-under-par 275. Matthew Jones and Dan Buchner both carded 67s on Sunday to end in a tie for sixth at minus-12.
Snedeker opened with a birdie on the first, but gave that stroke right back as he bogeyed the second. Birdies on the seventh and 10th got him to 15 under par, one stroke clear of Price.
Price carded birdies on the fourth and sixth, but faltered to a bogey on seven. He atoned for that mistake with a birdie on No. 8 and made it two straight as he also birdied nine.
The Australian converted back-to-back birdies from the 11th to get to 15 under and into a share of the lead.
Snedeker and Price each birdied the 14th, but Snedeker fell one back as he bogeyed No. 15. Price closed with four pars in a row to get in a 16 under. Snedeker drained a 15-foot birdie putt at the last to force the extra session.
'Aron played really well today. A 66 on the final day is really good,' said Snedeker, who became just the eighth 54-hole leader in 28 events to hang on for the win this year. 'Luckily I hung in there and played great golf down the stretch.'
Boo Weekley closed with a 6-under 66 to jump into a tie for eighth. He ended alongside John Merrick and Matt Weibring at 11-under-par 277. David Hearn, Ryan Armour and Kyle Thompson were one stroke further back at minus-10.