CHANDLER, Ariz. — The most automatic play in the Patriots’ offensive playbook over the last 14 seasons involves Tom Brady, but it is not a pass. It is having Brady tuck the ball and lower his shoulder and shimmy into a narrow gap, gaining inch after bruising inch, the quarterback with a fullback’s mentality.

Brady is as unstoppable as a crash of rhinos on third or fourth down and 2 yards or less to go, the rushing equivalent of a back-shoulder throw from Aaron Rodgers. Counting the postseason, Brady has run in those situations 115 times, according to play-by-play data from Pro Football Reference. He has gotten a first down or scored a touchdown on 105 of them, a success rate of 91.3 percent. Over one stretch, spanning more than seven years, he converted 60 of 61, including 37 straight.

Brady has joked that there is not a single cell in his body that tells him to run when a play dissolves and an opportunity to scramble presents itself. But the closer New England gets to the first-down marker or the goal line, the more Brady submits to his competitive makeup.

One of the Patriots offensive coordinator Josh McDaniels’s favorite such memories came three years ago, in the A.F.C. championship game against the Baltimore Ravens. A half-yard separated New England from a go-ahead touchdown in the fourth quarter, and Brady, scanning the defense, did not like the fourth-down play sent in from the sideline.