By Rob Hodgetts and Matt Slater
BBC Sport at Royal Troon |
The 31-year-old Englishman sunk vital birdie putts at the 16th and 18th, and a huge par putt at the 17th, to match the best round of the day.
Starting the final round on two under, Westwood appeared to be struggling when he sank to level par at the fifth.
But six birdies and no further mistakes carried him to within four shots of the deciding play-off.
Westwood was delighted with his storming finish, but said it could have been so much better.
"I'm very pleased, but I got off to a terrible start," he said.
"I haven't got off to a good start all week, which probably cost me the tournament." Westwood's previous best finish at an Open was also at Royal Troon in 1997, when he tied for 10th.
"I like the golf course. I think it's a great links test and they've set it up brilliantly," he said.
"I think it's as good as I've ever seen an Open Championship course set up."
Barry Lane dropped six shots in his final 11 holes to see his challenge collapse.
Lane had started the day at five under and within three shots of the lead but his final-round 75 pushed him back to one over.
The veteran had shot up to seven under after an eagle at the fifth, but things started to go wrong when he bogeyed the tricky par-three "Postage Stamp" hole.
Five further bogeys came in an inward 40 as his game fell apart under the pressure.
Paul Casey was happy with his overall performance after ending on one over following a final-round 72.
Casey, who led after the first round, achieved 20th position overall after a demanding four days.
"I can only profit from playing decent rounds on very, very tough courses. It should make everything else easier.
"I'm definitely becoming stronger mentally and that's good. I certainly feel I'm a better golfer than I was this time last year," he said.
Ian Poulter's golf failed to dazzle as much as his tartan plus-fours as he finished his Open campaign at two over.
The Englishman, who also wore a back to front flat cap, began at one over and promptly dropped shots at the first, third and fifth.
 |  It's only a matter of time before I win one |
A birdie on the seventh stopped the rot as he reached the turn in 38. The 28-year-old showed spirit to come home in 34 for a 72, but it wasn't enough for him to challenge at the end.
"I'm very happy," he said. "I've played really, really well. I've left a load of putts short this week but I've hit some great golf shots in tricky positions.
"I've proved to myself and a few others that I can play in majors and play well. It's only a matter of time before I win one."
England's Gary Evans trod water with a level-par 71 to stay at one over for the tournament.
The 35-year-old from Rustington dropped two shots in his first three holes but made amends at the fourth to go out in 37.
He leaked a further bogey at the 10th but picked up back-to-back strokes at 12 and 13 before parring his way home.
MMIX
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