The American, 22, is nine shots behind American joint leaders Keegan Bradley, Ken Duke and Charles Howell III.
"I got off to a poor start," said Spieth. "It was a very tough day."
Spieth, who will defend his Masters title in April, also missed the cut at the Northern Trust Open in February.
The defending champion never recovered from making five bogeys and a birdie in the first seven holes.
World number one Jordan Spieth is in danger of missing his second cut this year after carding an opening-round five-over-par 76 at the Valspar Championship in Florida.