Johnson built a 16-under par total of 197 at Club de Golf Chapultepec, where McIlroy fired a three-under 68 for 201.
It was Woods' worst weekend on Firestone's South Course since a 75-77 finish in 2010.
The trio made the most of perfect morning scoring conditions, finishing the second round on 11-under 129.
Poulter had eight birdies without a bogey to hold a one-stroke lead over Americans Kyle Stanley and Rickie Fowler.
His win comes with the start of the Masters, the year's first major event which he won in 2012 and 2014, looming on April 5 and Watson confident as he heads for Augusta National.
If he beats Watson on Sunday, Thomas will play in the afternoon for the crown against the winner of the other semi-final between Swede Alex Noren and American Kevin Kisner.
With two European Tour wins this season, Sharma has propelled to number 75 in the world, making his first start in an elite World Golf Championships event.
India's Shubhankar Sharma has won twice in the past three months, and outclassed a world-class field for 36 holes at Club de Golf Chapultepec near Mexico City.