The 26-year-old, who has signed a two-year deal with the League Two outfit, has been praised by Shrimpers boss Phil Brown.
Brown told the club's official website: "I'm delighted to have Myles joining us here at Southend United. His pace and trickery will cause defences in this division all sorts of problems.
"He knows what it takes to win promotion from League Two, having done so with Gillingham a couple of years ago. He's got that experience and should have a big part to play for us next season."
Weston was released by Gillingham earlier this month.