The visitors took an 18th minute lead when good work from Mason Mount on the right ended with a pass to Lawrence, whose low shot from inside the box took a wicked deflection on its way past Niki Maenpaa.
And the result was put beyond doubt on 77 minutes when a quick break out of defence saw substitute Mason Bennett feed Jayden Bogle, who fired into the roof of the net with only Maenpaa to beat.
City were reduced to ten men on 59 minutes when substitute Antoine Semenyo was sent off. The young striker was shown a straight red card by referee Darren Bond for a lunging challenge on Tom Huddlestone.
Derby now almost certainly need just one win from their last two games to finish in the top six.
Both teams lost central defenders to injury in the first half. Derby's Fikayo Tomori hobbled off on 11 minutes and was replaced by George Evans, who slotted in alongside skipper Richard Keogh.
City lost Nathan Baker on 33 minutes. Lee Johnson sent on Semenyo as substitute, moving Adam Baker back from midfield into his more accustomed position and switching Andreas Weimann to the right flank.
Derby's Kelle Roos was by far the busier keeper as City responded well to Lawrence's goal. Jamie Paterson posed a constant threat and was twice denied by diving Roos saves.
The Rams keeper also kept out efforts from distance from Semenyo and Josh Brownhill, while Paterson had another effort blocked, with Jack Hunt shooting wide from the rebound.
At the other end, the dangerous Mount just failed to get on the end of a Bradley Johnson cross and was also close to capitalising when Maenpaa palmed out a Martyn Waghorn cross.
Paterson began the second half with another jinking run. Roos made a fine save, but the ball remained in the Derby six-yard box and there was some desperate defending before it ran to Brownhill, whose effort was blocked.
Derby then had a great chance to make it 2-0 when Johnson broke onto a through ball inside the penalty area, but failed to beat Maenpaa, who spread himself to save.
Former Derby player Weimann cut in from the right to shoot just over. Then Jay Dasilva sent a 20-yard shot flashing past a post.
It didn't look like being City's day when, shortly after Semenyo's dismissal, Tomas Kalas had a header cleared off the line and Weimann saw a volley on the turn crash against the crossbar.
Mount shot wide when he should have settled it for Derby, but Bogle made no mistake and Frank Lampard's men had two goals ruled out for offside in the closing stages as City tired.