The match was marred by bookings for Reds goalkeeper Brice Samba and Lawrence as well as a straight red card for Ravel Morrison.
Forest are eighth in the EFL Championship table, two points off a play-off spot while Derby remain second from bottom on just 14 points.
Story of the match
The Rams had the opening shot of the match came in the ninth minute as Max Bird fired over the crossbar.
It was nearly two as Garner intercepted deep in Derby territory, the ball falling for Davis, who saw his chance saved by the foot of Allsop.
Substitutes combined for the visitors' next opportunity as Luke Plange fed Morrison, but his closed range shot was blocked and rolled out for a corner.
On 70 minutes, Allsop was nearly stranded as he came out of his goal, Grabban attempting a chip shot, but it drifted harmlessly wide.
The lead was doubled eight minutes from time as Johnson released Philip Zinckernagel and the Dane cut the ball back for Johnson to slide home.
Derby won a penalty after Cook fouled Lawrence and he stepped up to the spot, beating Samba in the bottom corner to cut the deficit in half.
Samba and Lawrence were booked in the aftermath and Morrison was shown a straight red card for a poor challenge on Zinckernagel deep into stoppage time.
Man of the match: James Garner
The Manchester United loaner was brilliant, tracking back on defence whilst showing off his full passing skills as he was seemingly everywhere on the pitch.