The clash was in the balance until Munster skipper Anthony Foley sealed the win by crashing over in injury time.
Despite fielding a international-packed side the Irish province failed to dominate the quarter-final and last season's runners-up crossed for two tries in 12 minutes during the first-half as they threatened to upset the odds.
Brendan Laney crossed for their first and hooker Dougie Hall burrowed his way over for the second on thehalf hour mark.
A blow to the Scots came in the form of a yellow card for lock Nathan Hines in first half injury time.
Munster's Alan Quinlan had opened the scoring in the 13th minute with a try after some good work from Lions lock Paul O'Connell.
Flyhalf Paul Burke added the extras but he missed with two further penalty attempts to give Edinburgh further inspriation.
In a swift counter attack Laney picked up and fooled his way past David Wallace and Anthony Horgan to happily dot down under the posts.
With Hall´s try, and two conversions from the recently-crowned Scotland player of the year Chris Paterson the score was 14-7 at the break.
An incisive try from Horgan soon followed, following some good work from Shaun Payne, and Munster had control of the game once more.
The Irishmen had further chances to wrap the game up but it wasn't until Foley barged over that the game was won.