Jimmy Gopperth and Charlie Hodgson exchanged penalties in the first half as the teams went into the break level at 9-9, Murray proving influential as the Falcons dominated the scrum.
But the game looked beyond the hosts in the second-half as Sale opened up a ten-point lead.
Sisa Koyamaibole crossed for the first try of the game on 55 minutes although it looked as though Marc Jones had knocked the ball on in the build up.
Hodgson added the conversion and then another penalty as the Sharks looked to ruin Murray's big day.
But Gopperth replied with a penalty of his own before Tim Swinson touched down under the posts - allowing Gopperth to add the extras to ensure Murray didn't end his debut empty handed.