Australian Scott picked up three shots around the turn to surge clear of his playing partner and overnight leader Snedeker who imploded with six bogeys before a brave birdie on the last hole gave him a round of 73 and a seven-under total of 203.
"It was a pretty good, solid round today," Scott told reporters. "I think 68 pretty accurately reflects the way I played. I certainly made a couple of nice putts.
"It's fun just to get in this position. It's what I've been practising for. I don't know what to expect tomorrow. I'm sure I'm going to be nervous but it's good nerves and I'm excited."
McDowell delighted the packed galleries with three late birdies to complete a rock-solid 67, the 2010 U.S. Open champion from Northern Ireland grinning broadly as he stayed in the hunt for a second major title.
"Since I was a young boy I dreamed of coming down that last fairway on a Sunday afternoon, the last group in the Open championship," said McDowell who was also in the final group at last month's U.S. Open where he finished joint second.
"I can draw on my experiences at the Olympic Club a few weeks ago. This is special for me and it was a lot of fun out there."
Woods, bidding for his 15th major, roared back after two early bogeys to card a level-par 70 and the former world number one finished in fourth spot on 204, one stroke ahead of three-times major champion Ernie Els and American Zach Johnson.
"It was not a very good start," said Woods. "I fought back and just didn't get anything going on that back nine. But even-par is about right."
Scott started with six pars and with his belly putter on fire, birdies at the seventh and eighth gave him real momentum.
The 32-year-old moved five shots clear with another birdie on the 11th and after missing a short putt to bogey the 13th he calmly parred the last five holes, helped by a brilliant wedge to escape from a bunker by the 17th green.
Snedeker, sporting a bright pink shirt, made his first bogey of the tournament at the fifth hole and his second followed immediately when he found a Lytham bunker for the first time.
He clawed a shot back with a delicate chip to set up a birdie at the par-five seventh but Scott matched him to move ahead.
Snedeker suffered his third bogey at the eighth and the Australian sank a monster putt to sprint three clear.
Clearly rattled, the American bogeyed the ninth as well after finding another bunker and Scott calmly made par to stretch his advantage to four.
Snedeker dropped further shots on 11 and 14 but rallied superbly with a birdie on the 16th and sank a 20-foot putt on the 18th to remain in contention for his first major title.
Woods recovered from bogeys at the first and third with a clutch of birdies at six, seven and nine to go to seven under.
A return to the metronomic golf which brought him two opening rounds of 67 led to five successive pars but a careless putt on the 15th green dropped him back to six under.
Els picked up two early shots to get to five under and after bogeying the 14th he made another birdie on 17 to stay in the hunt for a second Open title and fourth major.
World number one Luke Donald made an early birdie but took two shots to escape from a bunker next to the sixth green and the ensuing double-bogey stalled his progress and the Englishman carded a 71 to finish on one under.
World number two Rory McIlroy continued to struggle on the Lytham links with a scrappy 73 to finish five over and third-ranked Lee Westwood's wait for a major title will continue after his 71 left him at four over.
India's Anirban Lahiri caused the biggest cheer of the day with a hole-in-one, the first of the tournament, sending a nine-iron 150 yards into the cup.
"Just when I thought this experience was fantastic, it gets even better," said the 25-year-old after going round in 70 for a level-par total of 210.