Greig Laidlaw gave the visitors the lead after just six minutes with a well-taken penalty.
James Coughlan cancelled out Edinburgh's lead within a minute, scoring a try to put his side in front.
A second penalty from Laidlaw on 27 minutes cut the lead to just six points.
Three minutes into the second half, Conor Murray scored his first try of the game.
The home side's dominance continued without an answer from Edinburgh as Peter O'Mahony grabbed another try on 56 minutes, followed by a Keatley conversion.
Simon Zebo made it 31-6 on 65 minutes as Munster all but secured the victory with plenty of time on the clock.
Felix Jones added a late try on 76 minutes to round off the contest.