Werenski birdied three of the last four holes to sneak up on Thompson, who held a four-stroke margin until the 15th, where he missed an eight-foot birdie putt.
The pair started and ended the day sharing the lead, walking together to the clubhouse on Saturday evening at 15-under-par 198.
They ended two shots clear of South Africa's Charl Schwartzel, who had a strong day with the putter to finish with a five-under 66, and American Tony Finau (69).
Finau, the world number 17, struggled through much of the day at just one-under 14 until a birdie on the 15th took him within two shots off the lead.
England's Tom Lewis, the only Briton to make the cut, opened with a pair of birdies en route to an impressive four-under 67, which left him tied for 33rd, eight shots off the lead.