It left the 53-year-old a disappointing 17 over par for the tournament on his home course of Royal Troon.
Russell Knox (two over) was the highest placed Scot, with Paul Lawrie 10 over.
"I haven't really performed that well, but at the same time I've always enjoyed the whole theatre of the thing," said Montgomerie.
"I'd love to think this won't be my last Open but it might be."
Henrik Stenson hit an eight-under-par 63 to win The Open by three shots on 20 under after an enthralling final-round tussle with Phil Mickelson.
Montgomerie, who hit the very first shot of the 145th Open on Thursday morning, also started the ball rolling on Sunday at 07:40 BST due to him being bottom of the 81-man pile overnight with a total of 12 over.
Four bogeys and a birdie on the front nine meant the eight-time European Order of Merit winner's score continued to slide deeper into the blue.
Another two bogeys at the 11th and 12th holes did more damage before making par at every hole from the 13th to the 18th.  Montgomerie eventually finished 78th of the 81 players who made the cut.
"This whole year I've struggled," said the Scot, whose best finish at The Open is runner-up to Tiger Woods at St Andrews in 2005.
"I've struggled pulling the ball and the ball is going left. When you are aiming left, expecting a fade, you know, it's doubly bad. So I've just got to sort it out."
Montgomerie now goes on to play in the Senior Open at Carnoustie which starts on Thursday, but has not ruled out trying to take part in another Open.
He joined the field for the first time in six years after finishing in the top three at a qualifying event at Gailes Links in Irvine.
"You've got to be realistic and think that it might well be the last time I'm here at an Open," he said.
"The Gailes Club, part of the Glasgow Golf Club, were very hospitable and I thank them for allowing me to try and qualify to get here in the first place.
"I'd love to try and do that again."
Knox finished joint 30th after a two-under final round of 69.
He said: "I had two goals today. One was to par the second and one was to hit the 11th fairway and I checked both boxes.
"I played excellent today and I was really happy to finish with a good score. I probably finished where I deserved to finish."
Scotland's Colin Montgomerie posted a final-round five-over-par 76 in what he hopes is not his last experience of The Open Championship.