Chris Dean had opened the scoring for Edinburgh but Murray's double, the scrum-half benefiting from a dominant Munster scrum, and academy product Conor Oliver's try before the break put Munster in the box seat.
Dave O'Callaghan ensured the Irishmen also secured a try bonus point before the hour mark as Munster recorded back-to-back wins.
Back rower John Hardie grabbed a late consolation for the visitors but it wasn't enough after Tyler Bleyendaal had successfully converted all four Munster tries.
A number of injuries saw the game stretched to over two hours in length and Erasmus was delighted to come away with the full five points.
"I would never have wanted the game to feel like a horror movie, just watching minutes and minutes and minutes as it felt like we weren't getting there," the South African said.
"Especially the last 15, 20 (minutes) when we played with a prop at flanker, it was really getting long and frustrating. But yes, five points is great.
"The four tries we scored were nice but there were so many others we could have scored. There were simple handling errors and there were times when we had four-on-two when we ran into the player in front of us.
"Luckily, if you play this bad and win, and get five points, it's probably a good sign."
For Edinburgh, it was their third defeat of the season as yellow cards to Fraser McKenzie and prop Allan Dell severely hampered their cause.
They travel to Guinness PRO12 champions Connacht next weekend who have yet to win a game this season following their most recent defeat to Scarlets.
Guinness PRO12 Munster News
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