The hosts - sent into bat beneath overcast skies - have had to scrap hard against the only other side with a 100 per cent record in the North Group.
But they will be the happier of the teams after losing only two wickets in challenging conditions, England star Jonny Bairstow out for 22 and Tom Kohler-Cadmore dismissed for a fourth ball duck.
After handing a first-class debut to George Hill, the 19-year-old all-rounder who came in for off-spinner Jack Shutt from the side that beat Nottinghamshire at Trent Bridge last week, Yorkshire lost a wicket in the second over.
Kohler-Cadmore tried to withdraw his bat to a delivery from Ben Aitchison which he succeeded only in edging to Matt Critchley at third slip.
Adam Lyth pulled Michael Cohen for successive leg-side sixes in the third over, Cohen struggling during three unsuccessful overs from the Kirkstall Lane end before captain Billy Godleman hauled him off.
Godleman was the culprit with the score on 15-1 in the seventh over, the Derbyshire captain dropping Bairstow on nought at mid-off off a fiercely-hit drive off Dustin Melton.
It was a chance that should have been grabbed, and Bairstow rubbed salt into the visitors’ wounds by square driving Aitchison to the boundary, twice hammering Melton to the off-side rope and then pulling the same bowler for four.
But Bairstow fell with the total on 40 in the 16th over when he pushed at a ball outside his off stump from Anuj Dal which he edged to wicketkeeper Harvey Hosein, having faced 40 deliveries and hit four fours.
Lyth battled through the session for 25 and Dawid Malan is with him on 23.