Burkhead, who saw additional snaps in the absence of veteran tailback James White, had two rushing touchdowns and a receiving score in the win. White missed his second straight game after his father, Tyrone, was killed in a car accident earlier this month.
"We tried to go out there and play for him," Burkhead said of White. "He's a brother to me. I really feel his pain."
Sony Michel also stepped up in the Patriots' backfield, finishing with nine carries for 117 yards. As a team, New England rushed for a season-high 250 yards against the Raiders.
"We came together and just played hard," Michel said. "That's what James White does."
With Sunday's victory, Patriots head coach Bill Belichick joined George Halas (318) and Don Shula (328) as the only coaches in NFL history to reach 275 regular-season wins. New England (2-1) has won its last six meetings against the Raiders and improved to 47-9 in regular-season games following a loss.
Patriots star quarterback Cam Newton completed 17 of his 28 passes for 162 yards with one touchdown and an interception. Linebacker Shilique Calhoun forced two fumbles, including one that was recovered for a touchdown.
Derek Carr was 24 of 32 passing for 261 yards and threw two touchdowns for the Raiders (2-1), but he had two costly fumbles that led to a field goal and a late touchdown.
Raiders running back Josh Jacobs also had a fumble. He finished with 71 yards on 16 carries.
"We fumbled the ball in the red zone. We missed a field goal that we should make. We can't stop the run in the second half. There are some issues that add up to a loss," Raiders head coach Jon Gruden said.