Henley had back-to-back birdies at the seventh and eighth holes, and the 10th and 11th, before rolling in a 19-foot putt at the 13th in Norton, Massachusetts on Sunday.
The American - 62nd in the FedEx Cup rankings - closed out the round with another birdie at the 17th for a six-under-par 65 at the TPC Boston.
That birdie putt ensured Henley heads into the final round with the overnight lead at 12 under through 54 holes after compatriot Billy Horschel ended strongly.
Horschel, who is 82nd in the rankings, fired a third-round 67 after birdying his final three holes of the day to be perfectly poised.
Worrying for Henley and Horschel is the ominous form of world number one Rory McIlroy, who shot the equal-best round of the day to rocket up the leaderboard on Sunday.
After making a slow start to the second FedEx Cup play-off event, McIlroy is now only two shots off the pace after his seven-under-par 64.
McIlroy went bogey-free throughout his third round, with seven birdies helping the Northern Irishman into a share of third alongside American Chris Kirk and Australian Jason Day, who lost his spot atop the standings.
Kirk enjoyed an identical round to McIlroy, with seven birdies the highlight in his 64.
As for Day, he was left to rue yet another poor end to his round.
Day missed the opportunity to take the outright lead on Saturday after faltering with three bogeys in his final seven holes, including the 18th.
And it was a similar story on Sunday as Day bogeyed the 14th and 15th holes for a two-under-par 69.
American Webb Simpson (68) is in sole possession of sixth spot at nine under overall.
A shot further back are American trio Kevin Streelman (65), Keegan Bradley (69) and Ryan Palmer (71), and South Korean Noh Seung-Yul (68).