Centre Charlie Venables kicked the Yorkshire side ahead with a 12th-minute penalty, but their lead barely lasted a minute before the home side were off the mark with a try from Tom Vooght.
Five minutes later, Henry Bird claimed a second home try as a third came via Ollie Taylor with 34 minutes gone.
Venables added a second penalty shortly after half time, but the Tykes had work to do if they were to break through a resilient Rams defence.
Vooght scored his second try for Rams on 58 minutes, and another came on 74 minutes from Ollie Cole as the Rams exerted their dominance.
Leeds eventually broke through on 77 minutes with scrum-half Connor Lloyd touching down, but it was no more than a consolation.
Hull Ionians ran out 31-17 winners in a thrilling Yorkshire derby clash with Huddersfield in National Two North.
Huddersfield opened the scoring with centre Will Milner kicking an early penalty, but the first try went to Ionians with winger Josh Britton racing in after 15 minutes.
Wing forward George Mewburn added a second on 25 minutes, hooker Ben Stephenson claimed their third, and when Britton added his second on 41 minutes they already had a four-try bonus point in the bag at 26-3.
No 8 Lewis Bradley pulled back a try for Huddersfield on 74 minutes which Milner converted, but three minutes later Ionians’ No 8 Callum Sanderson restored the gap with their fifth try.
Centre Lewis Workman scored on 79 minutes and Milner converted, but time ran out for a comeback.
Trailing 17-0 after 20 minutes, Rotherham Titans hit back to win 49-19 at Blaydon and maintain their unbeaten start to the season. Two converted tries and a penalty gave the home fans something to cheer about, but it proved a false dawn.
Flanker Harry Williams dashed over on 20 minutes and wing forward Lewis Allan added another on 28 minutes, stand-off Joseph Carlisle converting.
Replacement Callum Bustin added another on 46 minutes, and Jonny West’s conversion gave them the lead at 19-17. Centre Harry Dunne was next on the scoresheet and after two penalties from West he added the conversion following winger James Cooke’s 73rd minute try.
Alex Pleasants was next on the scoresheet, West converting, and replacement Francis Entressengle claimed their seventh and final try. Hull emerged 31-10 winners from their home clash with Chester, claiming five points from the victory.
After the visitors had kicked a penalty in the first minute, Hull opened their account on 19 minutes with second row Cameron Curry scoring and stand-off Simon Humberstone converting.
Prop Michael Jobling scored another, and after a penalty try was awarded they led 19-3.
Chester pulled back a converted try just before the break, but Hull extended their lead after half time with No 8 William Hall and winger Keane Naylor scoring tries. Wharfedale are still searching for their first league win of the season following a 66-13 loss at Bournville.
Stand-off Jack Blakeney-Edwards kicked a penalty for Wharfedale and tries came from scrum-half Lewis Kincart and wing forward Chris Jackson.
Harrogate fared no better, losing 80-26 at Fylde.
Wing forward Matt Lansdall and second row Sam Brady scored tries in the first half hour, and stand-off Sam Fox added the conversion to keep them in the match at 24-14. However, it was mostly one-way traffic after that, Fylde opening up a 73-14 lead before Harrogate could add to their scoreboard.
Full-back Sam Parry and No 8 Fraser Kitching scoring late tries, but it was just a consolation.
Sheffield Tigers lost out 32-16 to Sedgley Park at Dore Moor, centre Jamie Broadley scoring their only try while stand-off Mark Ireland kicked the conversion and three penalties.