And that lead means Tiger Woods will have to make history to win his 15th major and deny the Englishman his first at the 62nd attempt.
Westwood carded an eagle and three birdies in a round of 70 that left him atop the leaderboard on three under par, with Woods (72) and fellow American Hunter Mahan (68) the only other players in red figures on one under.
All three men are coached by Canadian Sean Foley, but Westwood joked: "He's taken the chicken's way out and flown home to Florida I think. You won't see him on the range tomorrow [Sunday]."
Masters champion Adam Scott, who 12 months ago squandered a four-shot lead with four holes to play at Lytham, is three off the lead on level par, with Sweden's Henrik Stenson, Zach Johnson, Angel Cabrera and Ryan Moore another stroke back.
"I know what to expect tomorrow [Sunday]," Westwood said. "I know what to do. I know what it takes. It's just a case of believing you are good enough to win.
"When you analyse it, you don't want to say it, but tomorrow [Sunday] is just another 18 holes. I'm playing well and putting well and there's no reason why I can't carry on.
"I have had lots of chances. I could have won four with the right things going my way. They are the things you feed off. You try to learn from the things you did wrong and put them into practice.
"I'll think about winning the Open Championship tonight [Saturday] at some stage, I'm sure. I don't see anything wrong with that - picture yourself holding the Claret Jug and seeing your name at the top of the leaderboard. But when it comes time to tee-off tomorrow [Sunday] I should be in the same frame of mind as I was today [Saturday]."
Westwood also held the 54-hole lead at the Masters in 2010 and shot a closing 71, only to be overhauled by Phil Mickelson's final round of 67.
"I thought I played nicely that day at Augusta, but Phil played great," he added. "Sometimes you play well and someone plays better."
Several groups were put on the clock for slow play, including that of Woods and Westwood, but only Japan's Hideki Matsuyama was penalised a shot on the 17th, turning his par five into a bogey six and leaving the 21-year-old six off the lead.
Westwood led by three shots when he eagled the fifth and birdied the seventh, but saw that wiped out by the ninth and was tied with Woods after holing from 20 feet to scramble a bogey on the 16th.
But the 40-year-old then birdied the 17th as Woods bogeyed to take a welcome cushion into the final round, where he will play with Mahan and Woods with Scott.
Scott's caddie Steve Williams worked for Woods for 13 of his 14 major victories, all of which have come when he led or shared the lead going into the final round.
The last of those was in the 2008 US Open at Torrey Pines, where Westwood and Woods played together in the final round and Westwood missed out on the play-off between Woods and Rocco Mediate by a single shot.
Westwood has also come close to breaking his major duck in the Open, most agonisingly in 2009 when he three-putted the 72nd hole believing he needed a birdie, only to again fall short of a play-off - between Tom Watson and Stewart Cink - by a shot.
"I had a chance at Turnberry that I messed up a bit that I can fall back on, getting out of the zone and focusing too much on what other people are doing," Westwood added.
Woods had employed a conservative strategy over the first two days at a parched Muirfield and it looked to be paying dividends when he briefly moved into the outright lead with a birdie on the second.
He and Westwood then split the fairway on the 559-yard par five - Woods by using a driver for the first time all week - with Westwood then hitting his driver again for his second shot to leave his ball just short of the green.
The 40-year-old, putting superbly since working with 1991 Open champion Ian Baker-Finch, was faced with a testing putt from around 35 feet but promptly holed it for an eagle three.
"My emotions were pretty calm all day although when something exciting happened like that eagle I allowed that to pump myself up," Westwood said. "You don't really expect to make those but I could see the line easily breaking in off the hill."