The former England Under-19 midfielder moves to the MLS side on a free
transfer after his six-month contract with Donny expired.
"We have acquired a young, talented and versatile player with the potential of being an impact player in MLS," said Dynamo president Chris Canetti.
"Although he's only 24 years old, he brings a wealth of playing experience at some reputable English football clubs.
"This is a very exciting signing for the club."
Barnes has also previously had spells with Derby County and West Bromwich Albion.