Middlesbrough had not scored since early October but they should have gone ahead in the second minute when Derby lost possession in the penalty area only for Britt Assombalonga's shot to deflect behind off Matt Clarke.
It summed up Boro's recent luck in front of goal and Derby almost made them pay when Jack Marriott broke up the right before he was denied by a sliding challenge.
The game was very open and Derby unlocked Boro's defence in the 22nd minute with a quality move started by Bielik and finished with a low drive from just inside the area by Lawrence after he exchanged passes with Marriott.
Boro's afternoon got worse in the 32nd minute when Saville went in for a challenge with Bielik, catching the Polish midfielder just above the right ankle, and was shown a red card by Darren Bond.
It looked the correct decision and Marriott was only just off target with a stooping header before the visitors headed for the dressing room with a lot to sort out.
But Derby penned them back from the restart and should have scored again when Jayden Bogle cut the ball back to Chris Martin who blazed over from eight yards.
Duane Holmes dragged a shot wide as Boro were stretched again and the midfielder flashed another effort just off target in the 57th minute.
The visitors had created little but they had an opening in the 67th minute when a deep cross was headed back but Marcus Tavernier failed to make contact.
Jonny Howson rescued his side when he cleared a Bogle cross-shot close to the line and Pears had to claw away a Lawrence free-kick in the 75th minute.
But Derby made the game safe in the 84th minute when Bogle played in Lawrence, who showed good control before driving a shot past Pears from an angle to pile more pressure on Boro head coach Jonathan Woodgate.