His offensive coordinator with the Twisters (Ron Calcagni) recommended him to the Dallas Cowboys, who in turn would sign him in August 2001. Swinton made the team and set single-season club records for kickoff returns (56), kickoff return yards (1,327), and combined kick return yards (1,741). His 13.7-yard average on punt returns ranked third in the NFL and second in the NFC, and his 23.7-yard average on kickoff returns was ninth in the NFL and seventh in the NFC. Even though he was not given punt return duties until week 7, his 414 punt return yards were the seventh most in club history and tied the mark combined kick returns (87). This production earned him second-team All-Pro honors from "College and Pro Football Newsweekly" as a punt returner.