The Croatian stopper gifted Derby both goals, completely missing a Matty Pearson back pass before he was beaten at his near post by a Lawrence cross in the second half.
Lawrence, who started his second game following a drink-drive charge, was booed by sections of the home fans although the Wales international was cheered when he scored.
Bennett was not in the squad but Graeme Shinnie made his Sky Bet Championship debut, which meant he was facing his older brother Andrew, one of two changes for Luton who found themselves trailing after a howler from Sluga.
Derby started brightly, testing Luton with deep crosses from both flanks, but Pearson's back pass in the 11th minute should have been routine for Sluga only for the keeper to let it roll under his right foot and over the line.
It was an astonishing mistake and there was another at the other end when a low ball from the right presented James Collins with an open goal but he put it over the bar from virtually on the line.
Luton looked too open and Derby cut them apart in the 35th minute when Jamie Paterson easily slipped past Pelly Ruddock-Mpanzu and found Chris Martin, but his pass was just in front of the incoming Florian Jozefzoon.
Derby were also getting caught in possession and Collins had another opening in the 43rd minute but could not get his header on target as he stretched.
Lawrence, who was booed before the start, got in a good position early in the second half but dragged a shot wide and Luton had a chance in the 56th minute when Ryan Tunnicliffe fired over from 10 yards.
The visitors were having a good spell and Harry Cornick went close when his low ball was turned behind for a corner as the home fans became increasingly anxious.
Derby responded with a slick move that saw Jayden Bogle miskick before Lawrence cut inside and fired narrowly wide in the 63rd minute.
But Lawrence added a second in the 70th minute when his cross from the right sailed over Sluga, who was again badly at fault.