The current Scottish Amateur champion closed with rounds of 71-70 on the Old Course to get himself into a play-off with English pair Jack Cope and Robin Williams as they all finished with eight-under 280 totals.
Burns was bidding to make a triumphant return to St Andrews where he graduated with a first-class degree in economics, only to be denied by Cope as he prevailed in the shoot-out at the fourth extra hole.
English Amateur champion Cope, who had a spell at Stirling University, had earlier closed with a bogey-free 67, having birdied the fourth, fifth, sixth, 10th and 18th in a polished performance.
“Very much staying in the present,” said Cope of his last-day performance. “I know I’m capable of shooting two good scores to get into this.
“My caddie helped me out a lot as well, lots of motivational stuff, you know it in yourself but you also need to keep hearing it.
“I love coming here to the home of golf, I was at University in Scotland for a few years and I played all the courses here. I particularly like the Old Course, you just walk round and think who’s won here and played here.”
Cope, who is a member of The Players Club near Bristol, beat Callan Barrow, the new Scottish Men’s Open champion, in the final at Woodhall Spa last year to land his English title.
“I was committing to every shot and I maybe played the best golf I’ve played all week on those four holes. That was a nice time to do it,” he added, having driven the 18th green twice.
“I feel I know how to win, and having done that, and that (the English Amateur) being a big event, I wasn’t overawed by the situation.
“I’ve played four play-offs in my life and lost every single one. I was thinking my luck’s got to change sometime.”
Turnberry’s Ruben Lindsay also closed with a five-under effort to just miss out on the play-off along with Aucterarder’s Rory Franssen and Barrow