Thompson, who like Horschel is seeking his first win on the tour, shot a bogey-free, 3-under 67 over the Sea Island Resort's Seaside Course for a three-day total of 13-under 197.
Horschel, with whom Thompson was paired, birdied the first two holes but did not make another birdie all day. He made a 10-foot par putt on the final hole to salvage a 70 for a 198 total.
Thompson will take the lead into the final round of a tournament for the first time in his career.
Immelman, who shot a 62, and Simpson, with a 69, were tied for third at 199. Former British Open champion Louis Oosthuizen was part of the group tied for fifth at 201.
Simpson needs to finish no worse than 15th this week to vault past Luke Donald into the top spot on the tour's money list. Donald, ranked No. 1 in the world, is hoping to become the first player ever to lead the money list in both the United States and Europe in the same year.
Donald is not playing this week but has committed to take part in the season-ending event at Walt Disney World next week -- setting up a showdown with Simpson for the money crown.
Notable deaths of 2014 [PHOTOS]