The youngster signed a four-year deal at the Keepmoat Stadium, and has set his sights on helping Sean O'Driscoll's men win promotion to the Championship.
"I spoke to the chairman plus the manager and they convinced me that it would be a good career move," he told the Yorkshire Post.
"I want to play in at least the Championship and I know this is a good chance to move up with the club."
United had held talks with Woods, who was also said to be interesting Barnsley, Norwich City and Sheffield Wednesday.