Willett carded a third round of 72 in testing conditions at Augusta National to finish level par and a shot ahead of stablemate Lee Westwood, whose 71 made him one of only five players to break par in the strong winds.
At that stage the English pair were six and seven shots behind Jordan Spieth respectively, only for the defending champion to bogey the 17th and double bogey the last to throw the tournament wide open.
"I'm expecting that we're five or six back going into the last day, in with a reasonable chance, but you're going to have to do something special," said Willett, who only arrived in Augusta on Monday after the birth of his first child last week.
"This golf course, even if you're hitting it really well, you've still got to make putts on these greens. Obviously with crosswinds and fast greens, it's never easy.
"The golf course has firmed up day by day so hopefully if the wind does die (on Sunday), we can still keep the control of the ball flight and hopefully roll a few in."
Westwood, who was never outside the top 10 at Augusta from 2010 to 2014, looked out of contention after a front nine of 39, but stormed back in 32 with birdies on the 12th, 13th, 14th and 18th.
"It's tough when the wind is up. There's so many hard holes and not that many birdie chances, only really two, I suppose, for me, because I can't get to eight," Westwood said. "And you've just got to hang in there.
"You know it's possible around here to shoot four or five under on the back nine, so it's a case of not losing your patience and not being too aggressive when you don't need to be and then just taking the chances when they present themselves on the back nine."
Spieth admitted he would struggle to forget how his poor finish had left the destiny of the 80th Masters up in the air.
Spieth held a four-shot lead with two holes to play on a windswept day at Augusta National, but bogeyed the 17th and double bogeyed the 18th after wild drives to card a 73 and finish three under par, just a shot ahead of fellow American Smylie Kaufman.
''I just have to absolutely throw it away, the finish to this round, pretend it's a new round, everyone is tied and you have to shoot the best score to win," said Spieth, who is aiming to become the fourth player after Jack Nicklaus, Sir Nick Faldo and Tiger Woods to win back-to-back titles.
"(I have to) understand it's the position I wanted to be in after 54 holes and not think about the finish to this round. It's going to be very difficult. I played the last three holes, the last two days, five over par. There was no challenge in those holes really.
"If I'm at five, six under, that certainly brings anyone who is over par almost out of the tournament. And now with very little wind (on Sunday), someone gets on a run and shoots six, seven under, I know I have to shoot a significant under par round in order to win this tournament, when I could have played a different style of golf like I did on Sunday last year."
Asked how he would clear his head, Spieth joked: "Probably go break something really quick, have dinner and watch a movie. I think it will be tough personally. That wasn't a fun last couple holes to play from the position I was in."
Kaufman's 69 was the only sub-70 round since Thursday and secured his place in the final group with Spieth, whom he played against in junior and amateur golf - without much success.
"He's probably 1000-0," said the 24-year-old, who still lives with his parents despite carding a closing 61 to win his maiden PGA Tour title last year. "He's always beating me. Granted, he was so much better than I was as a junior and amateur. I was kind of a late bloomer in that regard. Just kind of kept on getting better."
Rory McIlroy suffered a hugely disappointing third round at Augusta National. He began the day just a shot off the lead but evoked memories of his final round collapse in 2011 as he stumbled to a 77, failing to card a single birdie for the first time in 81 major rounds.
Willett's fellow Sheffielder Matt Fitzpatrick is tied 23rd after rounds of 71 76 74.
He had four bogeys and two birdies in Saturday's third round Bernhard Langer is just two shots back of Spieth alongside Hideki Matsuyama and is in contention to become the oldest major champion ever by a decade, with world No 1 Jason Day, Dustin Johnson and Willett on level par.