The 26-year-old, who has also played for Crystal Palace and Luton Town, had been on trial with the League Two newcomers after leaving Vale at the end of last season.
Mansfield have not revealed the length of Andrew's contract.
He told BBC Sport: "I've kept myself fit over the off-season and have put in a lot of work to get myself right. I feel I did myself justice in the few games which I've played in."
Andrew made 24 appearances for Port Vale last season, scoring once.