Carr finished with 144 yards and completed 13 of 17 passes. His first touchdown was a 17-yard strike to Cordarrelle Patterson to give Oakland a 6-3 second quarter lead.
Amari Cooper caught the second touchdown of 48 yards to establish a 13-10 Raiders lead heading into halftime and finished with a game-high 78 yards on four catches.
Ezekiel Elliott played one series for Dallas, rushing six times for 18 yards and catching two passes for 6 yards. Elliott's efforts were part of a 12-play drive that started at the Cowboys' 26-yard line. Kicker Dan Bailey's 42-yard field goal gave Dallas a 3-0 lead with 6:05 left in the first quarter.
Carr's first scoring drive went for 10 plays and 81 yards, but a missed extra point gave Oakland a 6-3 lead in the second quarter.
Dallas responded with a six-play, 80-yard touchdown drive to take a 10-6 lead as Dak Prescott hit Jason Witten for a 10-yard score with 12:10 until halftime. Prescott completed 11 of 14 passes for 113 yards along with rushing twice for eight yards in the first half.
Oakland regained the lead 13-10 with Carr's touchdown pass to Cooper with 3:07 until the break.
Rookie quarterback Cooper Rush worked with the Cowboys' second team, completing 12 of 13 for 115 yards and two touchdowns. The two scores came after linebacker LaTroy Lewis recovered Rush's fumble in the third quarter with a 65-yard return to give Oakland a 20-10 lead.
Rush threw a 7-yard touchdown pass to running back Ronnie Hillman and cut into Oakland's margin 20-17 with 12:20 remaining. The Cowboys took control 24-20 with 7:02 to go when Rush found receiver Lance Lenoir for a 44-yard touchdown.
Oakland's Marshawn Lynch carried twice for 6 yards.
Witten caught six passes for 74 yards and a touchdown for Dallas. Cowboys running back Morris rushed for a game-high 56 yards on nine carries.
Dallas rookie defensive end Taco Charlton recorded his second sack of the preseason in the third quarter.