The 31-year-old former New Zealand international will be a straight replacement for Australian Clint Greenshields, who is leaving the French club after six years to return home.
Webb scored 86 tries in 156 appearances for the Rhinos, including four Grand Final victories and a World Club Challenge, since joining them from New Zealand Warriors in 2007 but he missed their recent Challenge Cup final defeat by Warrington with a knee injury which has also ruled him out for the rest of the season.
"I'm very happy to welcome Brent into our squad for next season," said Dragons coach Trent Robinson.
"He is a class player with a wealth of experience which will add value on and off the field."
Webb is the second confirmed signing by the Catalans for 2013 after the recruitment of Newcastle Knights forward Zeb Taia as a replacement for Setaimata Sa.
They are also being linked with Bradford second rower Olivier Elima, the France captain who is a former Dragons player.
- New Zealand Warriors