The 2012 champion was one stroke behind overnight frontrunner Bill Haas, and was still in the same position when he reached the beginning of Amen Corner.
Watson made par at the 11th, which is proving to be the most difficult hole so far this week, before going on to pick up shots at 12 and 13, where he reached the green in two and could have had an eagle.
Another birdie followed at 14 thanks to a bending 40ft putt, and a chip from the back of the 15th set up a short conversion for a fourth straight hole under par.
That would become five birdies on the spin at the 16th, although his round would end on a sour note with only his second bogey of the day coming at 18, failing to save par after a wild shot off the tee.
Watson, who won at Augusta two years ago after edging out South Africa's Louis Oosthuizen in a playoff, went bogey-free in a Major round yesterday for only the second time in 80 attempts.