The 25-year-old Streb had a 10-under 206 total. He won the Web.com Tour's Mylan Classic in September and finished seventh on the tour's money list to earn a PGA Tour card.
Horschel had a 68, also at Del Monte, as the field of 81 played in light morning rain and soft conditions in the tournament that features PGA, LPGA, Champion and Web.com tour players.
"I'm very excited," said Streb, who turned pro in 2009. "I didn't know what to expect coming out here. The greens were very soft and the ball wasn't traveling. In those conditions, you're just trying to get by."
Streb, who began the third round trailing the lead by two strokes, had four birdies and one bogey.
"I had no idea whether I'd play worth a darn of not," Streb said. "But it turned out to be pretty good."
Champions Tour players Russ Cochran and Kirk Triplett faltered after sharing the second-round lead. Cochran had a 74 at Spyglass to join William McGirt at 7 under. Triplett had a 75, also at Spyglass, and was 6 under.
"It was just too much of a day with the weather," said Cochran, the leader among the 11 Champions Tour entrants. "But I played OK. It was just kind of an uneventful day."
Cheyenne Woods, the niece of Tiger Woods, shot a 73 at Del Monte to finish at 1 under -- the best of the three women who made the cut for the final round Sunday at Pebble Beach.
Hall of Famer Annika Sorenstam had an 81 at Pebble, leaving her at 4 over. Juli Inkster, who in 1990 became the only woman to win the event, also was 4 over after a 75 at Spyglass.
The winner will earn $60,000 from the $300,000 purse.