The 40-year-old Westwood, one of the finest players never to have won a major championship, followed opening efforts of 73 and 71 by shooting a 70 to tuck in three strokes behind joint leaders Bubba Watson and Jordan Spieth on 214, two under.
The Englishman would have fared even better but for a stroke of misfortune on the 15th green when a curling 50-foot eagle putt touched the cup but somehow stayed out.
“I don’t know how that missed and I’ll probably watch that again and again,” Westwood told reporters after another roasting hot day at Augusta National.
“It was one of those things. I won’t tell you what I said. I can’t…for the lip readers watching.
“I feel like I know how to get around here even if I’m maybe not on my game and I’ve hit the ball well this week.”
Asked if his 70 had thrown him right into the title mix, Westwood replied: “Oh, very much so, yeah, yeah.
“Anywhere within five, even six shots of the lead going into the final round of the Masters is giving you a good chance especially with the way the golf course is playing out here.”
The former world number one said the greens were getting faster and firmer and the key to victory was to stay patient on the putting surfaces.
“The greens are kind of on a knife edge,” he explained. “I thought I hit a great shot into the last and basically had to grind to make a par four.”
Westwood, who has slipped to 37th in the world rankings, saw playing partner Brandt Snedeker slump to an eight-over 80 on Saturday.
“I watched arguably the best putter in the world today five- putt the fourth hole,” said the Englishman. “That’s the kind of thing we’re dealing with out here.”
Westwood’s compatriot Rose endured an up and down start to his third-round 69.
“I four-putted the second … and then the third hole was obviously a bonus,” said the U.S. Open champion after finishing on 215.
“I hit a driver off the tee and wedged it up. It’s a blind shot but I could tell by the crowd that it went in.”
Rose also made good strides up the leaderboard on the par-fives, eagling the 13th and birdying the 15th.
“On 13 I hit a great drive that drew around the corner,” he said. “I then hit an awesome four-iron at the pin and made a 25-footer.
“Being four back going into tomorrow is the biggest result that I could have looked for going into today,” said the 33-year-old Rose.
(Editing by Gene Cherry)