Geno Smith tossed a screen pass to Chris Johnson but Antonio Smith was able to hit him behind the line of scrimmage for a loss on the play.
Charles Woodson tallied the first interception of the season when he picked off Geno Smith halfway through the first quarter.
Carr dropped back and was able to find Rod Streater in the middle of the field for a gain of eight.
On 3rd and 2, Carr found James Jones on the outside for a short gain that resulted in a first down.
With just under three minutes left in the first quarter, Derek Carr threw his first career touchdown to Rod Streater who galloped 12 yards into the end zone.
Chris Ivory took a handoff from Smith but Charles Woodson met him in the backfield and was able to force him out of bounds for no gain.
On the final play of the first quarter, Geno Smith attempted to rush for a gain, but Carlos Rogers met him behind the line of scrimmage for a loss of two.
With the Jets on the verge of finding the end zone, Sio Moore laid a huge hit on Geno Smith that caused him to fumble the ball. Rookie TJ Carrie recovered the fumble.
Deep in his own territory, Carr tossed a beautiful back-shoulder pass to Streater for a gain of 16.
Geno Smith tried to connect with his tight end Jace Amaro, but Tyvon Branch provided the coverage and was able to force the incompletion.