Clement enjoyed back-to-back victories, his first since taking charge during the northern summer, in the two matches since the resumption of the Championship after the international break but the former Chelsea, Paris Saint-Germain and Real Madrid assistant has still not tasted success in four outings at the iPro Stadium, drawing three times.
His side dominated possession and regularly held the initiative but, until late in the game, never troubled Tom Heaton in the Burnley goal.
Burnley offered little as an attacking force but the point sees it climb to third place in the table, while Derby is up to 11th spot.
George Boyd's well-timed pass to Andre Gray presented Burnley's record signing with an early sight of goal 10 yards out but Richard Keogh got across to block his shot.
Derby had claims for a penalty turned down in the seventh minute when Chris Martin collided with the onrushing Heaton. Crucially though, Heaton had got a touch on the ball.
The Rams remained on the front foot for the rest of the first half but could not penetrate a well-organised Burnley backline.
Burnley might have taken the lead just before half-time when a David Jones corner created uncertainty in the Derby six-yard box and the ball bobbled about before Boyd hooked it over the crossbar.
Derby resumed after the interval where it had left off and Keogh powered a near-post header narrowly over the crossbar from a Johnny Russell corner.
But the story was largely much the same as the first 45 minutes as Clement's team, while enjoying sustained periods of pressure, never troubled Heaton.
When the Burnley goalkeeper was beaten by another Keogh header in the 77th minute Tendayi Darikwa was in the right place at the right time, as he had been for much of the match, to nod away the ball from just in front of his goal line.
George Thorne worked Heaton for the first time a minute later with a 25-yard shot that the former Manchester United trainee athletically tipped away.
Derby could sense a breakthrough might be on the cards but as it increased the pressure on Burnley the game became stretched and the visitor started creating chances with previously unseen regularity.
Sam Vokes stung the palms of Scott Carson with a 20-yard effort and then guided a Boyd cross, which was driven into the box with some force, wide of the target.
The Wales international then made Carson get down to his right to smother a shot from inside the area.
Back at the other end, Heaton had to repel a Martin shot from outside the box in the final minute that looked to be heading for the bottom corner.
