Westwood held a one shot advantage going into the final round at Augusta National but was eventually beaten by Mickelson, who closed with a faultless 67.
It was yet another near miss for the Englishman, who came close to securing victory at last year's Open Championship and US PGA Championship.
But the world number four is hopeful his time will come and revealed he had received some consoling words from Mickelson after the pair had finished their rounds.
"I have finished third-third-second in my last three Majors so I think I am doing the right things," said Westwood. "One of these days the door's going to open for me.
"Phil was saying in the scorers hut that he has been that man. He has hung around at the top of leaderboards at Major Championships and not got anything and there he is today sitting there with four of them.
"He told me to keep plugging away and that I was playing far too good not to win one of these.
"When I'm playing like this and contending for Major Championships I think I'm a lucky man.
"I think whenever you come as close as I've come here there's a tinge of disappointment, but that doesn't last too long."
Westwood was pleased with the way he finished the tournament, which ensured he moved ahead of third-placed Anthony Kim and maintained the pressure on Mickelson.
"I kept in it to the end," he added. "I thought I was unlucky at the 15th because I had a spike mark on my line and the ball hit it and went straight left.
"Then I hit a great shot into 16 and just missed the putt and two great shots on 17 (where he made birdie) so I played well down the stretch when I needed to.
"Any time anyone in the last couple of groups hits 67 in the final round they are probably going to win the tournament and Phil knows how to get it round here.
"He didn't play flawlessly all day but he has got such a magical short game that he can afford to hit it in the trees and get away with it.
"Then he played 12 and 13 fantastically and was solid coming in like a true champion."