The 39-year-old, who finished runner-up in 2010 and third last year, was in contention for the lead after birdies on the second and seventh moved him to three shots off top spot.
However, he found Rae's Creek on the 13th hole to finish three-under par and admits that he needed to do better on the par-fives.
"I missed my chances out there," Westwood told Sky Sports News. "I could have been five six under after eight and I was only two under.
"The par fives have cost me. I haven't played them well enough. There are not too many negatives to take away and I was on the edge of contention.
"I know my way around this golf course and I felt that I played quite well."
Westwood finished the week without a three-put in Augusta.