The pair battled it out in difficult conditions at the Stoop, with Freddie Burns' haul of 11 points making the difference, while Tom Brady scored the Tigers' only try.
A late Sam Simmonds touchdown under the posts made things interesting but the Chiefs were made to regret their profligacy with the boot, as the East Midlands side collected their third victory in the modern era of the tournament.
Assistant coach Murphy was full of praise for his side's determination on Sunday, especially after being forced to defend their own line during the closing stages.
Murphy said: "The guys have been passionate and committed throughout this tournament and some of the defensive shots that went in during the final quarter were fantastic.
"There was real ferocity in the tackle and it was some heroic defence at the end."