The Rockets' playoff hopes suffered a major blow and their loss also meant that the defending champion Dallas Mavericks clinched a post-season berth.
Houston lost its sixth straight and fell a full game behind Utah in the battle for the final playoff spot with three games to play.
The Rockets led by as many as 13 points in the first half but stumbled down the stretch of regulation.
Courtney Lee missed a free throw with 15 seconds to play that would have given Houston the lead and Landry then scored the first five points in the extra period.
Eric Gordon had 27 points for the Hornets and Al-Farouq Aminu had a season-best 17. It was the fifth New Orleans win in six games.
Goran Dragic had 23 points and eight assists for the Rockets.