Ben Jones-Bishop got the hosts off to a flying start when he touched down from Kevin Sinfield's kick after only three minutes. Sinfield kicked over the first of five goals to give his side a 6-0 lead.
Ryan Hall went over for the Rhinos on 11 minutes, before Louis Anderson got the Dragons on the scoreboard with a try soon after. Scott Dureau kicked over the extras to make the score 12-6.
However, Jones-Bishop bagged his second score of the match just minutes later to give Leeds a 16-6 advantage at half time.
Their lead was stretched even further after the break thanks to Zak Hardaker's score, but Steve Menzies's try on 51 minutes kept the Dragons in the match.
Two quick tries then put Leeds in control as debutant Shaun Lunt and Ryan Bailey both touched down.
There was late joy for Catalans in the form of Leon Pryce's score, but it was a mere consolation for the visitors.