Davis was on course to secure his first US PGA Tour title as he held a one shot lead at the halfway stage of the final round in Texas.
He had entered the day sharing the lead with America's Bryce Molder but the Briton birdied the first, second, sixth and seventh, with a bogey at the par four fifth the only blemish on his outward nine as he moved to 19 under par.
However, he was one over par on the back nine, bogeying the 18th, to allow Johnson to steal in for a three stroke victory, with his total of 21 under par setting a tournament record as he followed yesterday's six under 64 with the same score today.
The American had started the round a shot behind Davis and Molder but a stellar back nine 31 gave him the advantage.
Scotland's Martin Laird was back in the clubhouse after a final round 65 which included an eagle and four birdies moved him to 13 under par.
That was three shots adrift of American Ryder Cup Captain Corey Pavin.
Paul Casey finished with a four under par 66 to close at 12 under, alongside Korea's K J Choi.