The hosts dominated the opening exchanges and took a deserved lead as Max Hayman dotted down from the back of a maul, Mike Cooke’s conversion attempt from the right falling short in a strong wind.
But Leeds hit back having gained territory with a long penalty clearance which earned them 40 metres.
A slick move off a line-out on half-way followed with Connor Walker diving over next to the posts for a score added to by Ed Crossland.
Zach Clow was then yellow carded for taking an opponent in the air from the restart, but the numbers were level early in the second quarter as Tykes No. 8 Eddie Brown infringed in his own 22.
And Rams then replied as the industrious Harry Stone powered over from close range.
Their opponents regained the lead after turning the ball over in midfield, however, Seramaia Bai Turagabeci crossing for a converted score and 14-10 lead.
Rams pull clear either side of break
Kalling-Smith was then held up over the line as the hosts responded again, the Swedish international successful second time round to nudge his side back in front shortly after the half-hour mark.
The bonus-point was then secured before the interval as Zach Clow’s sweet sidestep saw him touch down on the left, Andrew Lamb with a superb conversion for an eight-point advantage.
The lead was extended early in the second half with Kalling-Smith finishing a brilliant team score featuring plenty of strong carries and sublime off-loads with a walk-in effort on the left.
Cooke was unable to add the extras, but he was on target after Kalling-Smith completed his treble with a marvellous solo effort in clubhouse corner.
With Rams now 20 points up midway through the third quarter the home fans would have wanted them to cut loose, but Leeds battled gallantly and came close to reducing the deficit when Hayman did brilliantly to hold the ball up.
Leeds pick up bonus-point
The Yorkshiremen did, though, keep their hopes alive as the impressive Albert Bradshaw crossed on the left.
The final 20 minutes proved rather tepid, however, albeit the visitors had the last word and secured a try bonus-point as Walker burrowed over following a spell of sustained pressure.
Rams remain eight points off the play-off places ahead of a trip to Birmingham Moseley next Saturday (3pm).
Final Score: Rams 34–24 Leeds Tykes (Half Time: Rams 22–14 Leeds Tykes)
Scorers
Rams
Tries: Hayman 6 Stone 22 Kalling-Smith 32, 43, 49 Clow 37
Conversions: Lamb 38, Cooke 50
Leeds Tykes
Tries: Walker 13, 72, Turagabeci 27 Bradshaw 55
Conversions: Crossland 14, 27
Star Man: Axel Kalling-Smith (Rams)
Referee: Sam Yates
Attendance: 723
Rams
(15-9, 1-8) Lamb, Clow, Jones, Grundy, Kalling-Smith, Cooke, Hoadley; Marris, Hayman ©, Lomas, Elston, Jackson, Stone, Duda, Bicknell
Replacements: Marshall, Harris, Baker, MacRae, Hodgson
Leeds Tykes
(15-9, 1-8) Davies, Bradshaw, Dixon, Crossland, Krippner, Bai Turagabeci, Laughton; Pleasants, Davis, Hough, Eley, Guthrie, Walker, Teague, Brown
Replacements: Yorke, Williams, Bell, Green, Keith
