The Swede bogeyed his first hole, the tenth, and his penultimate hole, the eighth, but sank seven birdies in between to lie in a tie for third alongside Canada's Mike Weir, one stroke behind leading duo JB Holmes and Davis Love III.
Holmes needs a top-four finish in Florida if he is to have any chance of securing himself a place inside the world's top 50 and an invite to the Masters Tournament and he got off to the best possible start with a six under par round of 66.
The 27 year old birdied his first hole, the tenth, and claimed an eagle at the par five 12th. He produced further birdies at the fifth, sixth and ninth as he completed a flawless round.
Love shot seven birdies and an eagle on the way to his 66, a shot ahead of Stenson and Weir.
Next up was a group of four players at four under including Ernie Els and Kevin Streelman, Kevin Na and Robert Allenby.
While Holmes was setting the pace it was not proving to be such a promising opening day for some other players in need of results if they are to head down Magnolia Drive in a couple of weeks' time.
England's Justin Rose lay three over par after double bogeying at the 13th, with the 29 year old requiring a top-four finish to have any chance to reaching the top 50.
Canada's Stephen Ames, who needs a top-five placing, was one over after his opening 18 holes while Bubba Watson was four over as he goes in search of a top-two finish.
Korea's K J Choi, who needs to finish in the top 55 to keep his Masters hopes alive, was tied for 31st at one under.
Northern Ireland's Graeme McDowell was the top British contender also at one under par while European Ryder Cup Captain Colin Montgomerie was at level par with fellow Scotsman Martin Laird back on two over.