The PGA Tour competition Valero Texas Open is a genre rehearsal for the Masters which is played at Augusta National and starts on Thursday.
44-year-old Stenson will be the only Swede in the season's first major, but he has not shown any further form.
Stenson was on the right side of the qualifying line before the last hole where he went on three bunker visits, which in the end resulted in a triple bogey.
The Swede finished with 73 strokes and was a total of four over par, which was not enough to make the cut.
The bang was just the latest in a row for Stenson, who for a couple of weeks was noted for the worst round of his career when he went around 85 strokes in The Players Championship.
The recent heavy results have caused the Swede to plummet in the world rankings.
Stenson, who is in 109th place, has now been passed by Swedish colleagues Alex Norén (98) and Henrik Norlander (105), which means that he can also miss the Olympics in Tokyo this summer.
If the gloomy development continues, future participation in majors is also in danger.
After the victory in The Open 2016, Stenson has been automatically qualified for all majors, but the benefit is only valid for five years and expires after the season.