The damage was done inside the opening quarter when Rhinos conceded three tries to go 16-0 behind.
Rhinos – without 10 players due to injury or suspension - staged a rally either side of the break to cut the gap to just four points, but they fell away again and the hosts completed a deserved win.
Leeds will take some consolation from the way they played at the start of the second half and Ash Handley and Richie Myler both had good moments in the game.
But Catalans had three tries disallowed – all correctly – and the margin could have been even worse.
On current form, Rhinos will struggle to hang on to their place in Betfred Super League’s top-eight and if they do manage that, they will be no more than also-rans over the final seven matches.
Rhinos made too many errors, conceded some costly penalties and their defence fell apart when put under heavy pressure.
Greg Bird sent Benjamin Jullien over for the opening try as Leeds made their customary poor start. Benjamin Garcia crossed from Remi Casty’s offload and Michael McIlorum touched down from acting-half, Josh Drinkwater converting the frist and third.
Tony Gigot had a try ruled out by referee James Child and his video assistant Chris Kendall before Catalans prop Sam Moa was sin-binned for a shoulder charge on Leeds’ ex-Dragons half-back Richie Myler.
Rhinos took advantage of the extra man to get points on the board when Ash Handley made a break with Myler.
He kicked to the corner and Handley touched down, though Jordan Lilley could not convert.
Gigot booted a drop goal to make it 17-4 at the interval.
Rhinos scored two tries in 13 minutes at the start of the second half to give themselves a fighting chance.
First, Brett Ferres ran a nice line on to a pass from Myler and then Ryan Hall finished strongly after Handley gave him the ball.
Lilley converted Ferres’ try and at only three points adrift Leeds had the scent of something. Catalans’ Leeds-born winger Jodie Broughton was fortunate to avoid a card when his high tackle prevented Jack Walker scoring a try which would have edged the visitors ahead and that was a key moment in the game.
Lewis Tierney had a touchdown disallowed by the video referee, for an obstruction on Ferres, but Gigot dummied over on 65 minutes and two later Drinkwater crossed.
He converted both to end Rhinos’ fightback and Dragons applied more punishment when Broughton scored their sixth try five minutes from time. Leeds had the consolation of scoring the final try, after Tierney had one disallowed for a knock-on.
Myler grubbered between the posts on the final play of the game and Brad Dwyer touched down, Lilley adding the extras.