Vincent had been released by Cheltenham Town, but the 28-year-old has wasted little time in securing himself a new club after Mellon moved quickly to conclude a deal for a player who he believes will fit straight into the first-team squad.
The 42-year-old told the club's official website: "He's someone that I've been aware of since I was at Fleetwood and he's a really fast, exciting attacking player with great technique.
"He can play on both wings and down the middle. He will fit right into the way that we want to play at Shrewsbury and I'm delighted to get him on board."
Vincent scored twice in 21 appearances for Cheltenham last season.