The 21-year-old Texan is at 11-under 205 and will enter Monday's final round trailing Paul Dunne, a 22-year-old amateur from Ireland, Australian Jason Day and 2010 British Open champion Louis Oosthuizen of South Africa by a stroke.
Spieth, who won the Masters and U.S. Open earlier this year, will be chasing a trio of players as he goes after the third leg of the Grand Slam.
With victory on Monday, Spieth would join fellow Texan Ben Hogan as the only players to win the Masters, U.S. Open and British Open in the same year and then head to next month's PGA Championship attempting to become the only player to win the modern day Grand Slam.
"At this point it's free rolling," Spieth said. "I'm going to play to win -- I'm not playing for a place. I don't want to place third tomorrow. I want to win.
"I'm going to play my game, obviously with patience, to stay in the mix if it's not all there in the beginning. And if it is, I'm going to continue to play that way to try and get out in front. Just trying to give myself as many chances as I can."
Dunne, who shot 66 during the third round, played college golf at the University of Alabama-Birmingham. He is first amateur to lead the British Open after three rounds since the great Bobby Jones in 1927. Jones was the last amateur to win The Open in 1930.
"It's surreal I'm leading The Open, but I can easily believe that I shot the three scores that I shot," Dunne said. "If we were playing an amateur event here, I wouldn't be too surprised by the scores I shot. It's just lucky that it happens to be in the biggest event in the world. Hopefully, I can do it again tomorrow. But whether I do or not, I'll survive either way."
Day and Oosthuizen, who won The Open at St. Andrews five years ago, shot 67s.
"I'm very happy with how I played today and I'm pleased to be in the lead," Day said.
Spieth began the day five strokes behind first-round and second-round leader Dustin Johnson, who struggled Sunday with a 3-over 75 to finish at 7-under.
Spieth made three straight birdies at the 10th, 11th and 12th, saved par from off the green at the 13th, birdied the 15th and made another save par putt at the 17th.
"I'm very pleased with putting myself in position," Spieth said. "Really happy to at least get back into it."
Padraig Harrington, who captured the Claret Jug in back-to-back years in 2007 and 2008, shot a 65 to move to 10-under and two shots back.
"I always wanted to shoot 65 on Sunday at an Open," Harrington joked in the clubhouse. "Unfortunately there's another round to go."
After rain and high winds delayed play for almost 14 hours on Friday and Saturday, the third round was pushed to Sunday and forced the oldest championship in the world to a Monday finish for the first time since Seve Ballesteros of Spain won at Royal Lytham & St. Annes in 1998.
NOTES: Zach Johnson was a paragon of consistency Sunday in the third round of the 144th British Open. He parred the first 13 holes, then went birdie, par, birdie, bogey, birdie for a 36-34--70 to finish 2-under and a 54-hole total of 9-under 207 and in a nine-way tie for seventh. ... Zach picked up five shots on the other Johnson, Dustin, who dropped from second-round lead into a group at 209. "I didn't hole any putts," said Dustin Johnson, who made only one birdie, at 15, and closed with three straight bogeys. ... Phil Mickelson, the 2013 champion, shot 70 for 212. "I had chances," he said. "We all had chances. You know it's obvious I didn't have the round today to get me in it for tomorrow, but I still appreciate being able to play here." ... Adam Scott, who let a lead slip away in the 2012 Open, said he was slightly off his rhythm Sunday shooting 70 for 207. "And that's frustrating because it was there to be had. But I'm still in the mix." ... David Duval hasn't had much success anywhere since winning the 2001 Open, but Sunday he shot a 5-under 67 for 211 and 5-under for through three rounds. "I expected to play well at some point," said the 43-year-old Duval, "but competing against the best golfers in the world, is not a fair fight when I haven't played a golf tournament in three or four months and they've been playing constantly."