Calcavecchia, the co-leader after the first and second rounds at Walton Heath, was in cruise control at 10 under with six holes remaining before imploding when his accuracy off the tee deserted him.
A disastrous triple-bogey 7 at No. 13 — after driving into the heather beside the fairway — was followed by another dropped shot on No. 16, although the 1989 British Open champion salvaged something from his round with a neat birdie at the final hole.
Calcavecchia, bidding to become the fourth player to capture the British Open double, shot 72 to put him at 7-under 209. His stumble gave a bunched-up field renewed hope in the Champions Tour's third major of the year.
Frost shot a bogey-free 66, tied for lowest round of the week on the parkland course just south of London, while Cochran's 67 included a double-bogey 6 on No. 14.
Corey Pavin, the United States' 2010 Ryder Cup captain, and England's Barry Lane each shot 69 and were a shot off the lead, with three more Americans — Chip Beck, Mike Goodes and Lee Rinker — at 5 under.
Three-time winner Tom Watson was one of four players a further stroke behind after a 69 in a third round played in overcast conditions with a light breeze.
Calcavecchia, who has been in a three-way share of the lead after every round this tournament, described his errant driving on the back nine as "army golf," because it alternated between going left and right off the tee.
"Today was exactly what's been happening to me all year," said Calcavecchia, who is without a tournament victory in 2011. "I drive it great for a while and then I just lose it. That's what I was worried about — and then it happens to me. I have to figure it out by tomorrow.
"I lose faith in myself. I could hit 20 perfect drives in a row and then hit one dead left and one dead right. ... I can't regroup. I can't find it."
Frost, who went out 10 groups behind Calcavecchia at 1 under, birdied the first, the last and four other holes in between in a consistent round.
"I made a lot of crucial putts. I had a good feeling for the speed and the reads today," said Frost, who revealed he "almost fainted" during the week, stumbling into a wall after standing up after dinner.
"I've got myself right in there now. Another round like this would be great."
Frost's 66 matched that of England's Kevin Spurgeon on Friday.
Cochran continued his impressive comeback from a wrist injury, sustained during the second round of the Liberty Mutual Legends of Golf tournament in late April, that ruled him out for two months.
He returned at Pebble Beach two weeks ago for the Nature Valley First Tee Open, where he finished tied for 10th, and he is within a shout of his first major title here after making seven birdies in his opening 11 holes.
Cochran also hit the pin with his 4 iron off the tee at the par-3 No. 17.
"I just had a great feel with the putter," said Cochran, who partnered with Beck (69). "We just fed off each other. It was a wonderful day."
Lee Rinker (74) and Canada's Rod Spittle (75) shared the overnight lead but couldn't maintain their form, although they are two of 12 players within three shots of the lead.
Eight of those are Americans, including Watson.
The eight-time major champion, who bogeyed No. 18, came off the course after his second straight round in the 60s looking in a regretful mood. At that point, he was six shots off the lead held by Calcavecchia, before his compatriot's late collapse.
"I had five lip-outs in a row starting at No. 7. I wish it was a bit lower ... but anything can happen on this golf course, with this heather," Watson said.
John Cook (72) and Peter Fowler (73) were also 4 under.
Defending champion Bernhard Langer of Germany was 1 under after a blemish-free 69, with his three birdies all coming on the front nine. He could only par his way home, denting his outside hopes of claiming the $310,000 first prize.